You mean just any structure that was built before 410 and is still occupied today? Keep in mind that ancient ground level is a dozen meters below modern ground level. If you go to the Basilica di San Clemente al Laterano, you enter in an 11th-century church, then go down a long flight of stairs into a 4th century level, then descend another flight of stairs into a Republican-era series of rooms. Rome is filled with such rooms, now buried under two millennia of subsequent building and fill. Most are not easily accessible and none of them are "occupied" normally except by thousands of rats and cats, intrepid homeless, and curious municipal workers.

Many modern apartment blocks in Rome seem to follow the same footprint as what was built in the Imperial period. Virtually all of them have sealed/buried Imperial-era rooms beneath them, but they are not typically functional or accessible. The buildings atop the were not built with access to older, lower layers in mind.

I wish I could remember the name of the presenter. Within about the last six months I saw a documentary on Rome, and in one part a presenter stood inside a tenement building, built with arched brick ceilings, which had been in continuous use from the Roman period until the 1930s. I can't find any reference to it online. They also talked about how the ground level had changed and why, and I seem to remember a hill that was a grassed over spoil heap of discarded pottery.

Maybe not quite what you mean but there is the Theatre of Pompey. It was dedicated in 55BC (Cicero talks about the opening games there in his letter to Marius: Fam.7.1.3.). It was the first stone theatre in Rome. The curve of the theatre audience bit, known as the cavea, was thought to have been about 157 metres in diameter. In its place is now a block of apartments that follows this original curve. Although the current building isn't considered the original, it is said that the theatre's foundations are in the building's basement, hence maintaining the curve. If you Google "Via di Grotta Pinta" you can see the curved building. At the centre of the cavea was a temple to Venus Victrix. It was thought to jut out like a figurehead on a ship might (Richardson 1987 p.126). Again not the original wall but the current building is built on the temple's foundations. If you go to the, think it's East end of the, Campo de Fiori you can see an apartment building come to a sharp sudden point and this is where the temple sat. (Richardson 1987, 'a note on the architecture of the theatrum pompei in Rome', AJA 91, p.123-126).

There's also the temple of Hadrian which I know less about. Dedicated in 145AD/CE (Historia Augusta, Pius, 8.), it now stands in the Piazza di Pietra and has been built into a building which now houses Italy's chamber of commerce. It's a really weird sight having this modern building sandwiching these huge ancient pillars.