Probably... though in that case, because you have OR, it won't really use both (unless you have index_merge and v5)... also, why are these left joins and not inner joins?
Basically, you need to give the server an way to find the matching row in the next table. You start with searching the entire forums table. You take the last_post_id, and look it up in the forum_posts table on post_id -- that should use the PK index. Then you need the two FKs in the topics table. Then the user_id (again, the PK). I'd like to see the current EXPLAIN for that 4 table query.