Well, depending on your definition, the last pharaoh of egypt was Cleopatra VII Philopator (yes, that Cleopatra; the last in the line of Greek Ptolemaic Dynasty established after Alexander the Great's death).

Cleopatra had several children,

• Caesarion, the son of Caesar. Likely died killed by Octavian, but there were no known records.
• Alexander Helios, the first son by Mark Antony. Thought to have died, killed by Octavian.
• Celopatra Selene II, the daughter of Mark Antony. Married to King Juba II of Numidia. Celopatra Selene had children with Juba, so it's possible there are descendants today that traces to her line.
• Ptolemy Philadelphus, the younger son by Mark Antony. Fate unknown.

Now let's look at Rome. The Western half canonically ended with Romulus Augustulus, who was young when deposed. There were no known issues. Romulus might have been spared with an annual pension granded by Odoacer, but this is uncertain.

So, of ancient Egypt and (western) Rome, there's only really one possibility that's not impossible to prove: descent through Cleopatra Selene.

Edit: Let's do more.

Byzantium (or the Eastern Roman Empire): Last emperor was Constantine XI Palaiologos, who tore off his imperial ornaments when the city fell and likely died in a last charge with his soldiers. He had no children.

The Ottoman Empire. Since Mehmed II declared himself Kaysr-i Rum (Caesar of Rome) upon conquering Constantinople, it's only fair to include them. The last emperor, Mehmed VI had many children. Ertugrul Osman was one of the issues who only recently died (in 2009). He lived in a modest NY apartment after the exile of the Sultan's family. I can't find out if he had any children, but he appear to have married twice according to wikipedia.

Muscovy (the state, not the city) made a claim to be the Third Rome through the marriage of Sophia, the niece of Constantine XI, to Ivan III. Its last emperor, Nicholas II, were executed by the communists along with the rest of the immediate Romanov family. There has been many claimed survivors.

The Holy Roman Empire (neither holy nor roman... yada yada) was dissolved by Emperor Francis II following the defeat by the French at Austerlitz. He had many children. So it's entirely possible (and more than likely) that there are descendants.

In a meaningful sense (i.e. "I'd be next in line for pharaoh if that were still a thing"), no, probably not. No one has cared, for some time, about the power that comes with that position, and we've stopped keeping track of who would have a legitimate claim. Likewise, as /u/cosimothecat points out, some of these people who were last in the line had no children, or we don't know what happened to their children, etc. This assumes that what we want to know is whether someone can claim to be a descendant of the last pharaoh or Roman emperor, and therefore "next in line." I think this is rather less fun, and the other answer will probably greatly disappoint anyone trying to make that claim.

In a sense that's more fun, the answer is surprisingly binary. For people who lived more than about 1000 years ago, if they have no descendants in a given population, then no, no one can claim they're a descendant. If they have any descendants in a given population, then every member of that population can claim they're a descendant. So, yes, probably every European can say they're a descendant of a Roman emperor.

By the way, they've looked at the Pharoah Ramesses III's DNA and found which Y-chromosome type he has. He is y-chromosome haplogroup E-V38. Anyone with that DNA may or may not be his direct descendant--there's no way to tell. But it means they share a direct male ancestor in common (that is, their father's father's father's etc. is the same).

By direct descendant, do you mean only in the male line? If that is the case then no, such titles have died out. If you however also include the female line then there are large numbers of people who can claim descent from at least the Byzantine Emperors. Although not common, it wasn't unheard of for Byzantine Emperors to marry some of their daughters off to Western Europeans, especially Holy Roman Emperors. Through the female line, many of Western Europe's nobility, such as the Queen of England would therefore be related to some of the Byzantine dynasties.

Tracing anything further back becomes very difficult though. Both the Byzantine and Roman period had numerous usurpers and general-emperors who were not related to the imperial line. Trying to connect some of the later Byzantine dynasties (e.g Komnenos) to current royal families is not particularly difficult, but it becomes much more difficult to connect the Komnenids to e.g. Constantine I or any of the Roman Emperors. Of course, many Byzantine imperials would indeed try to trace their line back to such figures (as the Doukases did for example), but there is no historical verification for such claims, and it seems much more likely to be nothing more than a fantastical claim.

One can imagine the process becomes even more difficult when it comes to the Pharaohs. Unless i'm mistaken (and please do correct me if you believe it to be so), female Egyptian royalty, unlike their male counterparts, could not marry below their rank, nor marry foreign royalty. This was done to ensure that the amount of claimants to the throne was limited. This makes it next to impossible to to trace any lineage from the New Kingdom to the Late period. The last Pharaohs were however not of Egyptian origin, but instead of Persian (Achaemenid) or Macedonian (Ptolemaic) origin. The end of the Ptolemaic dynasty has already been described by others, and the Achaemenid one ends in similar fashion, with the death of Alexander the Great's children. At this stage we are so far back in history it becomes very difficult to trace any kind of royal inter-marriage between kingdoms and empires. It's all within the realm of possibility that e.g. some Achaemenids ended up in marriages with more minor Macedonians, who in turn ended up in marriages with Romans, but it wouldn't be the ruling class any longer, plus there aren't any historical documents to verify it.

Ramses II was famous for being legendarily fertile, historians and archaeologists have found documentation for at least 90 sons and 60 daughters in various tombs and temples built during or shortly after the life of Ramses II (some 3.3k years ago).

That's nearly Genghis Khan levels of fathering children so it shouldn't be too surprising to think some direct descendants are still around. Sadly Egypt is a land that has known foreign invasion and occupation all too often since the glory days of Empire, and countless temples and burial site built to commemorate the dead have been lost.