Like a lot of questions about history, this is both superficially straightforward and on closer reflection highly philosophical. I have a very straightforward answer for you, one that I have never seen referred to in print or pixel before, but let’s take the complex route first.
We’d need firstly to define what we mean by Roman Empire. The Ottomans, the Germans, the Venetians, the Bulgarians, the Russians and a whole host of other civilisations all considered themselves in one way or another to be true heirs of Rome. Should we consider them as true continuations or not?
Then we’d need to consider what we mean by where. Where is a subset of when in this instance. If we define the Roman Empire as ending with the sack of Rome by Alaric the Visigoth in 410 CE, then obviously the empire fell at Rome itself. But Alaric was dead within a few months, and there was still an emperor in Rome over 60 years later.
And what of what we now call the Byzantine Empire, but which knew itself as the Roman Empire? Founded in Anatolia in the fourth century due to a split of the Empire into Eastern and Western administrations, the empire based in Byzantium (later Constantinople, later Istanbul) continued until it was overran by the Ottomans in the fifteenth century just as its Western twin was overrun by the Ostrogoths in the fifth.
For me, as for themselves, the Byzantines were Roman. Greek-speaking, yes, but Roman all the same, with a continuity of culture all the way back to the founding of Rome as a city state in the eighth century BCE. So if we consider the Byzantines to be the last vestige of the Roman empire, then surely it fell when Constantinople was captured by Sultan Mehmet II (known understandably as ‘the conqueror’) in May 1453 CE?
Close but not quite. Some of the Byzantine empire still stood even as Constantinople was sacked and burned. One standout was the Maniot territory in the Peloponnese in Southern Greece, which at the time Constantinople fell was under the command of the wonderfully named Despotate of Morea, which in practice meant two Byzantine leaders (who promptly fled.) The Maniot people did not flee however, and the Ottomans didn’t bother invading this mountainous and difficult territory until 1770 CE. But with the departing despots so also departed any vestiges of ruling Byzantine (and hence Roman) culture. This was a Maniot defiance of Ottoman rule, not a Byzantine one.
Then there was the principality of Theodoro, which was a sliver of Crimea under Byzantine rule sandwiched between the coastal Genoese colonies and the inland Khanate of Crimea. Technically again, this was Byzantine territory. But in reality, it was populated by Goths.
What? Yes, in fact the Ostrogoths had been in Crimea for over a thousand years, since the FOURTH CENTURY CE! Byzantine rule (following the fourth crusade) was merely yet another imperial vassalage for the Goths of Crimea. At various times they had fallen under the nominal rule of a bewildering range of imperial powers, including the Huns, Khazars, Mongols and Genoese. Ultimately, they were merged into the neighbouring Khanate in 1475 CE, and became part of the Ottoman empire. So, not exactly the last stand of Rome.
Which brings me to my own answer to the question, where did the Roman Empire end? The Empire of Trebizond was a secessionist state of the Byzantine imperium. Formed during the fourth crusade as an opportunistic power grab by a local potentate, the Trebizond empire sustained only a little longer after the fall of its parent state at Constantinople. The Trebizond secessionists were if anything even more aggressed by the combined threat from Turkmen and Ottoman forces than the Byzantines were. Throughout the 1440s and 1450s, they repelled repeated attempts at invasion.
The end finally came in 1461, a mere eight years after the fall of Constantinople. There is a wonderful, almost contemporaneous painting depicting the departure of the Byzantines from Trebizond following King David’s surrender to Mehmed II:
So what happened exactly? Mehmed swooped in from the west to isolate Trebizond and place it under siege, which continued for a month. To achieve this, his forces had to go into the high hills immediately behind the coastal city and outflank it, so that they would be unable to receive either reinforcements (which David hoped would come from Christian Europe) or supplies via the harbour.
Trebizond was a high walled city located between two freshwater sources flowing into the Black Sea, so a physical attack was ill advised. For Mehmed, it was easier to maintain negotiations while besieging the city. And the inhabitants were well aware of what had happened to Constantinople for refusing to negotiate.
This map, take from Wikipedia, gives a good sense of the geography of the time:
The formal surrender would of course likely have taken place in the citadel or the palace (both currently under archeological exploration at the time of writing.) However, this followed an agreement between David and Mehmet for a negotiated surrender. With their forces primarily located to the east of the city, adjoining the freshwater river that is now only a dry river valley in the modern city, it is possible that Mehmet’s forces first entered the city via the lower gate closest to the harbour and market, but more likely that they entered through the double gate closer to the citadel.
Amazingly, this gate is still standing, entirely unremarked upon, and can be found down a narrow cobblestoned alleyway strewn with graffiti and with children’s laundry drying at head height. There is no plaque or commemorative item of any kind to inform you that this place was the geographic spot where over 2200 years of continuous Roman culture came to its final end. And yet, that’s exactly what it is:
The inner gate of Trebizond’s double gate, where 2,200 years of Roman culture came to an end.