In Suetonius's "Twelve Caesars" he mentions in the Augustus chapter:
“18. About this time [Augustus] had the sarcophagus containing Alexander the Great’s mummy removed from the mausoleum at Alexandria and, after a long look at its features, showed his veneration by crowning the head with a golden diadem and strewing flowers on the trunk. When asked, ‘Would you now like to visit the mausoleum of the Ptolemies?’ he replied, ‘I came to see a king, not a row of corpses.’ ”
And in the section on Gaius (Caligula):
“52. Gaius paid no attention to traditional or current fashions in his dress, but ignored masculine and even human conventions. Often he made public appearances in a cloak covered with embroidery [....] He even dressed up as Venus, and long before his expedition he wore the uniform of a triumphant general, often embellished with the breastplate which he had stolen from the tomb of Alexander the Great.”
Excerpt From: Suetonius. “The Twelve Caesars.”
Do we know where the tomb was, and roughly when historians lost track of it? Have people tried to find it since? Do we have a guess as to what happened to it?
The latest mention of the body of Alexander comes from 215 under the emperor Caracalla. According to contemporary historians Cassius Dio and Herodian, Caracalla was obsessed with Alexander and had a strong desire to emulate the great conqueror. Herodian reports that Caracalla went to the tomb in Alexandria, opened up the sarcophagus, and placed his purple cloak and other assorted valuables inside. He then went back to his regularly scheduled business of massacring the Alexandrians because they weren't treating him with the respect he felt he deserved.
That is the last historical record of someone visiting the body of Alexander. What happened to it afterwards is a mystery. It is possible that the site was destroyed during Aurelian's campaign of 273 or that simple coastal erosion has consigned the tomb to the waves. However, mentions of a site alleged to be Alexander's tomb feature in later reports of Arab writers Ibn Abdel Hakam (A.D. 871), Al-Massoudi (A.D. 944), and Leo the African (sixteenth century). As mentioned in a piece on the subject in a publication for the Archaeological Institute of America (see here)all these accounts mention such a place but do not give its exact location. We can be reasonably certain that the tomb was in Alexandria and not Memphis, where it may have been originally following the body's arrival in Egypt in 321 BC. Ptolemy II (or perhaps Ptolemy I) moved the body to Alexandria in the early third century and the body was probably moved again later on by Ptolemy IV.
Despite the mystery that pervades this subject, there has been no shortage of effort expended on finding the tomb of Alexander. The medieval accounts mentioned above may well reference the real tomb and plenty of people have spent a lot of time and money trying to find it. Some 140 officially recognized attempts to find the site have been mounted over the years but with no success. A popular local legend states that the tomb lies below or near the Nebi Daniel mosque but it's not clear what evidence, if any, this supposition relies on. It may well be little more than nineteenth century romanticism. A team of Polish researchers working in the 1960s determined that the network of tunnels underneath the mosque were ancient cisterns rather than any kind of tomb. People still believe the legend though and subsequent, but fruitless, excavations have since taken place at the mosque. Other researchers of lesser repute have posited various alternative theories about the tomb's location but with little solid evidence.
The sum total of all this is that we don't really know what happened to Alexander's sarcophagus and tomb after the visit of Caracalla in 215. Coastal erosion may well have swallowed up the site but the fact that the location is so uncertain leaves us without a definitive answer. It may have been destroyed by Aurelian's war in 273 but we cannot say for sure. It is plausible that some or all of the later accounts refer to the real tomb and that it was still around by as late as the 16th century but specific details are lacking. Lots of people have looked for it without success but this is no deterrent for the most determined. No one has found any record of Alexander's earlier resting places either. In any case, Alexander's body has been an item of intense interest to a lot of people for over 2300 years (a war was fought over its custody after all) and this interest shows no sign of letting up.
For more see the article referenced above and The Quest for the Tomb of Alexander the Great by Andrew Chugg.