The omission of words from a sentence is known as ellipsis.
Whether or not the final verb can be omitted depends on what you think the complete sentence should be:
He doesn't like the movies he used to like.
He doesn't like the movies he used to do.
In my opinion, the first sentence is the best complete sentence and, as like occurs in the second clause in exactly the same form as in the first, it can be omitted from the second clause.
If you think that the second sentence is better, you cannot ellipsize do because it is a proxy for like in the first clause. It does not exactly match anything in the first clause in either form or function.
If the original sentence were
He doesn't do the things that he used to do
The do in the second clause can be omitted, because it exactly matches the do in the first clause.