Even before Caesar became a cultural icon for Renaissance Italy and unification, people liked to tell stories linking him to various things. Most of them are ridiculous. There are some stories that claim that Caesar's body was moved from the Campus Martius (where bodies were traditionally burned) to the forum by an angry mob, but it's not attested by anyone except Appian, and the episode bears far too much resemblance to the burning of Clodius Pulcher (memorably recounted by several sources) to be taken without a grain of salt. In any case, our ancient sources are pretty much all in agreement that his ashes were contained in the Temple of Divus Julius, built at great expense by Augustus, who wen pretty far out of his way to build up his relationship with the Caesarians by stressing that he was the dutiful heir.

What happened later? We don't know. There are plenty of stories of various barbarians destroying traditional Roman artifacts and stuff, mostly for rhetorical effect. We can't know generally whether those episodes took place or not. So after that, it's safe to say that at some point--probably when Rome was sacked some time or another--they were lost.