EDIT: Before reading my post and the rest of the comments in this thread, I would like to emphasise one thing to readers: In the period the OP is referring to, the Romans were not a race, they were a huge citizen body defined by culture but not by racial appearance. Please keep this in mind. /u/Celebreth's post below provides succinct and helpful clarification on the matter of 'ethnicity' in the Mediterranean then and now (ie. people look about the same) for those that are particularly concerned with this aspect of the question.

I understand how you would ask this question with the assumption that 'Romans' and 'barbarians' were mutually exclusive things, divided along racial lines by violent confrontation. However, Romanitas had less to do with ethnicity and a lot more to do with one's cultural affiliation. One could be culturally affiliated with both Rome and barbarian groups, and the fluency of the relationship between what was Roman and what was barbarian as evidenced by great 'Roman' characters of the late empire such as Flavius Aetius, Stilicho, Alaric the Goth (previously a Roman-appointed officer), or Arbogast (a Frank who served as a Roman officer and reached the rank of magister militum only to die fighting against the future Emperor Theodosius, half of whose army were Goths...). Italian aristocrats were still a politically and economically dominant group in the Empire, but it was a long time past the point where they could jealously regard themselves as the only Romans.

'Barbarians' were fully integrated into the Roman army and therefore the state itself, even in the highest command positions, in the late empire (Stickler 509). Large barbarian populations were settled and integrated into the Western Empire well before the mid-to-late 5th century AD (510). As late as the 4th and 5th century the Roman army was integrating and incorporating large populations of Frankish immigrants in Gaul to both the army and society, and likewise with large numbers of Burgundians in southern Gaul. These same Franks and Burgundians later "styled themselves...as descendants of the Romans (Amm. Marc. 28.5.11- Stickler 510). The catastrophic weakening of the Empire on top of its many other structural and situational problems by civil warfare in the last century or so of its political existence caused the many barbarian groups that existed in and around the ailing Empire to expand their influence and boundaries, and eventually the barbarian allies saw no choice but to join the Empire's carve-up in the face of its physical weakness. Many Roman military units, based on barbarian troops, "faded away" rather than being destroyed as the shift of power away from the Roman government in the West became more and more apparent: when the Western Roman emperor ceased to exist as a legitimate institution in the late 400s, these same once-loyal barbarians "lost a crucial point of reference connecting them to Ravenna and the empire" (510).

The barbarians that lived in the West in many ways could be considered "preservers" of the Empire rather than its destroyers: even after the Roman government collapsed and barbarians no longer served it directly, the major barbarian powers established new states in the vacuum. They "relied on the advice of Roman aristocrats and the work of Roman bureaucrats" and in many ways preserved Roman culture and as much of its institutions they could salvage (Bennett 26). Roman culture survived for much longer than the institution of the Western emperor. In the absence of a better description of the state of affairs in the immediate aftermath of the Western Empire's 'fall', I will quote Bennett at length here:

If we had a time machine in which to travel to Gaul in the year 500, we would find an exciting mix of peoples and places- rough barbarian warriors jostling with effete, Latinate aristocrats; once-grand cities shrinking in size and influence; Christian missionaries striving to convert both heathen barbarians and Arian ones; and in the countryside, a resettlement that sometimes mingled old and new and sometimes pushed out old families in favor of newcomers. This late antique melting pot created the European Middle Ages...it was mostly a synthesis of three factors...Roman traditions, Christian religion, and barbarian custom" (27).

So Romans did not go extinct, as they were not an ethnic group. In general, Romans were neither "exterminated" nor "displaced" but rather slowly assimilated into the Germanic societies and polities that settled and dominated Western Europe. For many of these 'Romans' they did not have to go far to assimilate, such as the Franks who already lived in Gaul and who as a society had rendered great service to the Roman government in the past. As such, the question of whether racial groups such as Italians, Spaniards, the French and "others" does not take into account the non-racial nature of what it was to be a Roman. Italy continued to be populated by 'Roman' Italians and their descendants just as it was settled by great numbers of Germanic peoples, Gaul continued to include the 'Romanised' descendants of Celts, the comfortably settled Franks and Burgundians, and many other Germanic groups besides. Spain must likewise have continued to have been populated by the descendants of 'Romanised' Iberians and Celtiberians just as it was settled by large numbers of Goths. Continued migrations in the Early Medieval period and phenomena such as the Viking invasions and Muslim conquest of Spain would have made further changes to this 'racial' melting pot. The modern Italians, Spaniards and Frenchman are therefore groups that have only become homogenous in the 1200ish years since significant immigrations (in terms of social impact at least) ceased into Western Europe, and their descent from 'Romans' would be just one aspect of the racial melting-pot that eventually formed into these modern 'ethnic' groups.

I think the complexity of this issue should also inform the difficulty of answering the question of whether or not 'Romans' were "Caucasian": Italians themselves were of varying ethno-genetic origin, some central Italian, some Celtic, some Greek, and many such mixtures in between. They would have had different complexions but must been broadly similar to modern Mediterranean people. Germanic and Celtic people of course were more pale, more classically 'Caucasian'. If you're a white person, you're probably more 'Germanic' than 'Roman', if that's what you're asking. But the real point I suppose is that 'Romans' were not a race. They were a culture, and the racial origin of those making up the Roman citizenry was everything from Celtic to Greek to Italian to Semitic to Syrian. Such a hodgepodge could never 'go extinct', but the culture and government from which 'Roman' drew its importance as a distinction could and did decline in dominance, or 'go extinct' in the case of the government.

Sources:

Bennett, Medieval Europe: A Short History, 2011.

Stickler, "The Foederati", from A Companion to the Roman Army, 2011, 495-514

Let's take these in order:

Did the Romans go extinct?

With the dissolution of the Roman and Byzantine empires, you could make an argument that "Romans" as such as extinct (though present-day inhabitants of Rome might disagree with you), but of course descendants of the population of the Roman Empire are still extant. According to The History and Geography of Human Genes 295:

Most of the groups that came to Europe after the fall of the Roman Empire were probably too small numerically to give a detectable genetic contribution, given that Europe was already fairly densely populated at the time.

Which leads into your next question:

Were they displaced or exterminated by the Germanic tribes?

You have the movement of multiple peoples into and around Europe during the "Migration Period" (c.400-700 AD), including Germanic tribes like the Goths, Angles, Saxons, Franks, Lombards, and Vandals. However, compared to the native population they were relatively small, and while they set up or took over kingdoms, they tended to be absorbed rather than displace or exterminate. A really good overview on the subject is Barbarian Migrations and the Roman West, 376-568, but the upshot is that when, for example, Odoacer declared himself King of Italy in 476, he didn't empty out Rome or put all the citizens to the sword - he reigned with the support of the Roman Senate.

To what extent are Italians, Spaniards, Frenchmen and others descended from Romans?

Long before the Migration Period, Rome expanded its borders from Britain to Northern Africa in the west, and Armenia to Egypt in the east. The Romans didn't have the numbers to displace the native populations any more than the Migration-era peoples did, but there was some intermarriage. To refer back to The History and Geography of Human Genes 281 on Gaul/France:

In Roman times, the population numbered about 5 million people, a high density for that time (though roughly 10 times less than now). The Roman language (Latin) fully replaced the Celtic dialects, of which only a few words survive in languages spoken today. Modern French evolved during the Middle Ages from Latin to this area, as summarized afterward in this section.

Later genetic contributions were probably scarce. German contributions include some German soldiers in the Roman army who were authorized to settle in France as farmers or as auxiliary soldiers. Visigoths were allowed by Rome to settle in Aquitania in A.D. 410, but later moved to Spain and left only a small genetic contribution. Burgundians, originally from the southeastern Baltic, settled in eastern France, but were crushed by the Huns. They gave the name to Burgundy, but survivors settled in Savoy. Another Germanic people, the Alemanni, settled in Alsatia and Switzerland and gave their language to these regions. Yet another Germanic tribe, the Franks, moved from the lower Rhine to north-eastern France, which was more sparsely settled at the time. The Franks are also believed to have contributed relatively little genetically to the Parisian area, which was then already heavily settled by Gallo-Romans, but they developed efficient political control of the region. [...] Altogether the German genetic contribution is estimated to have involved 300,00-250,000 people, or 6%-7% of the population of Gual (Fierro-Domenech 1986). Howeaver, Celtic people were themselves a mixture of central Europeans, and the high homogeneity of central Europe and northern Europe was probably already established, at least in great part, by late Neolithic times.

So if you want to define Romans as "inhabitants of the Roman Empire" - probably everybody in Italy, Spain, France, and most of Europe in general is descended from them. If you want to narrow that down into "descended from someone that was ethnically Roman," that gets trickier. Which leads us to your next question.

Were the Romans even caucasian?

"Caucasian" was one of the two categories of the human race (along with "Mongolians") created by Christoph Meiners in Grundriss der Geschichte der Menschheit ("The Outline of History of Humanity," 1785). It wasn't originally a scientific term, though it became adopted into various later racial typology schemes during the heyday of scientific racialism. So the exact answer would probably depend on how you meant "Caucasian." If you meant "Were the Romans white by the standards we understand it today?" then the answer would be: some of them! As I mentioned, the Roman Empire at it's greatest extant was hugely broad, and included a very ethnically diverse population. Certainly there were populations in North Africa and the Levant that were not what would be considered "white" today, and yet they were Roman citizens and intermarried with the general Roman population. You might want to read Lloyd Thompson's Roman Perceptions of Blacks, which is a bit dry but tries to show that attitudes towards race were a bit different.

I hope that helps answer your question.

There's a lot of debate, it seems, as to whether the Romans were ever a definable ethnic group (also, always a controversial subject for various reasons). Some answers may be found in Tacitus, who bemoans the decline of the ethnic Roman aristocracy (i.e. that which, according to legend, was formed from the mix of Trojans, Latins, Sabines and various Etruscans, and was then extended to other Italians). He fingers Claudius, in particular, for overruling the general conservative will of the nobles, to grant senatorial positions to Germans. Claudius himself made the argument, familiar today, that there was no reason they should deny the Germans this privilege, as were not most of the senators themselves descended from once excluded and enemy tribes? Tacitus seems to suggest this argument was intelligent, but really a Claudian scheme to increase his wealth through expansion while others warned that such behaviour may threaten Rome's existence (it did...eventually).

A variety of scholars in the 19th Century (for whatever that century's views on race may be worth..), including Nietzsche, make reference to the people of South Western France (around Bordeaux) being the most direct descendants of the ancient Roman lines, due to various convoluted and speculative assumptions about being enough out of the way of events to have survived as an ethnic group, among other things. Nietzsche proposes Guy de Maupassant as a classic 'Latin' man.

It seems certain that modern Italians would have some strong Roman 'bloodlines'. We know Italy is one of the countries in Europe with a stark haplogroup divide right down the middle - the southern part being almost entirely 'J', in common with Greece, North Africa and the Levant, among others, while northern Italy shares a group with Germany, France, Spain and most of the rest of northwestern Europe. This, as far as the general people, and not the noble classes of Rome, are concerned, is probably a similar situation to 2,000 years ago, as Romans were indeed a large and varied body of ethnicities, more so as time progressed.

On the question of roman DNA, hasn't some sarcophagi survived in tact and uninterrupted?