I don't think we know this for certain. None of the possibilities have been solidly confirmed. There is a lot of speculation every way. Some think LTT was never real and disappeared when Rand realised this. Others think that LTT was a real voice and that he simply integrated with Rand (since they were two personalities in a single soul). Or something.

We don't even know what caused LTT and the memories to appear in the first place. It could be the taint causing madness that let Rand's past life to bleed into the current one like Rand in the books believed. Or it could be the pattern somehow messing about to give Rand the skills required to survive against the forsaken.

We don't even know what is going to happen next. It does seem likely that Rand will have access to LTT memories partly or fully now, in a way similar to the way Birgitte had access to her past lives without any voices. This is just speculation however, it could be that with LTT apparently gone, Rand has lost all of his AoL knowledge and channeling skill, but that seems unlikely considering how big a role they have played in the books. I certainly would be disappointed if that happened.