Are there any ancient tombs (let's say no later than fall of western Rome), preferably still intact, that are known or believed to have belonged to significant ancient figures? Inspiration for the questions stems from curiosity of the tombs of Hannibal, Alexander, etc.
Those two that you mention we do not know where they rest. Well, we know where they used to rest, namely, somewhere in Bythinia for Hannibal (he has very humbly buried when he died on exile) and in Alexandria, Egypt, for Alexander. However, both places are long gone.
Besidds them, the resting places of some "famous" people are located nowadays: in Rome you can still visit the Mausoleum of the Scipios which, while doesn't hold the most famous of their kin, for Africanus was buried in his own villa, is still the burial site of many notable figures of the family. Also, there is the Mausoleum of Augustus, where the Princeps and a large part of his family were buried -cremated, but that's where the ashes were put-, and the same goes to the Column of Trajan and Hadrian's Mausoleum.
Outside of Rome, I recall three sites:
The first one is the royal necropolis of Aigai, the burial place of the Macedonian Royal House up until, at least, Philip II's father. It is possible that Philip himself and his son, Arrhidaios, successor of Alexander, were also buried there, but as far as I know this is not confirmed yet.
The second is known as the Tomb of Philopappos. Located in Athens, in a hill close to the Acropolis, there is a large monument that used to keep the remains of a certain prince Gaius Iulius Antiochos Philopappos, grandson of Antiochos IV, the last king of Commagene. He was the first born son of the heir apparent to this kingdom, a small principality, client of Rome, in what is today eastern Turkey, in between the 1st and 2nd enturies AD. However, they were eventuaƱly deposed by the Empire and allowed to live as high ranking private citizens. After this, Antiochos Philopappos lived peacefully in Athens, where he became a prominent citizen, and, after his death, his sister Julia and the citizenry of the city funded this huge monument in his honor.
Lastly, there is the least known of them, the Belevi Mausoleum. Located in western Anatolia, near Ephesus, it allegedly held the remains of king Antiochos II Theos, lord of the Seleucid Empire between 261 and 246 BC. While in a really bad state, the mausoleum must've been, judging by its size, on a similar level to that of Halikarnassos. Information about it on the Internet is scarce, but, apparently, a stone sarcophagus was found inside, housed now in a museum in Ephesus.
I'm sorry I can't be of any more help- I'm afraid there are not that much places we now of where such important people where buried that you can still visit. Hope this helped, though!