We often think of everything between the Roman Empire and the Renaissance as "The Dark Ages", hence the name Renaissance. The question then is, did the people living in these so called Dark Ages realize that there was such a thing as the Roman Republic with pseudo-democracy, or the Roman Empire? What did they think of it? I of course realize that only a very small percentage of people were literate, so knowledge of the Republic would hardly reach the peasants to cause an uprising.
Literate medieval people not only knew about the Roman world, they were steeped in it, if for no other reason than the educated language of medieval Europe—and far beyond—was Latin. Roman ideas about philosophy, politics, art, literature so permeated medieval high culture that it’s hard to answer your question easily or briefly. So, medieval culture has long been described as a melding of three earlier cultural influences: Roman, Christian, and Germanic. From the Roman perspective, scholars had access to all the major writers of Roman antiquity and were well aware of the history of the Republic and Empire. They knew the histories of Livy (Dante cites him as “Livy, who cannot err”), Tacitus, Suetonius, and Sallust, to name a few, the miscellanies of Aulus Gellius, the rhetorical works of Cicero (widely admired) and Quintilian, the natural science books of Pliny the Elder and Younger, and the major literary works of Virgil (likewise widely admired—remember, he escorts Dante through Hell), Ovid, Horace, Terence, Seneca, Juvenal, and some Martial and others. In fact, one estimate claims that Carolingian scholars of the 8th-10th centuries preserved some 90% of the Latin canon as we know it, so much so that when Renaissance book-hunters went scavenging through medieval monastic and cathedral libraries, they mistook works copied by Carolingian scribes in their innovative minuscule hand (the precursor of the alphabet you’re reading now), for original Roman copies of the works they were trying to recover, imitated it, and thereby passed it on to us. The Carolingians, remember, thought they were revitalizing the extinct western Roman empire when Charlemagne allowed himself to be crowned Roman Emperor in Rome in 800 though for his circle the imperial heart had been transferred from Rome to his palace at Aachen. Roman culture flourished in court circles among people like the scholar Alcuin who was deeply grounded in Roman works. Carolingians even imitated Roman cultural styles. For instance, the main arch gateway of the monastery at Lorsch was inspired by Constantine’s Arch in Rome.
In literary terms, there were three major sources for inspiration for medieval writers: the matter of Britain (namely, the tales of King Arthur and his court), the matter of France (famously represented in the chanson d’geste The Song of Roland), and the matter of Rome, featuring Greek and Roman mythology and tales of Alexander the Great and Julius Caesar. In early Christianity, there was unease about adopting Roman culture, which, on one hand, seemed antithetical to Christian values. The great church father Jerome, who adored Roman culture since he considered himself part of it, related a dream where, standing before Jesus, he claimed “I am a Christian,” and was challenged that he was in fact a “Ciceronian” because “where your treasure is, there will your heart be.” But his contemporary, the even greater church father Augustine, helped sanction adopting many aspects of Roman culture. In fact, in his widely known, gigantic City of God he passed on many aspects of that culture.
As for Rome’s political influence, that’s a vast topic but anecdotally it’s worth mentioning Cola di Rienzo (d. 1354), who tried to restore the Roman Republic in Rome, and declared himself both a Tribune and a Senator and actually had a popular following for a few years until the mob turned on him and murdered him.
A great source for exploring the influence of Rome on the Middle Ages is Grafton, Most, and Settis’ The Classical Tradition (2010) arranged by topic, mostly by authors.