Before you even say "I forget" (for the case when you can't remember his name at that moment), the time of you "forgetting" has already past. His name isn't -going- from your mind that moment; it's -gone- out of you mind at that moment. Therefore, I think that the only time you should say "I forget" is when you regularly can't remember something.
The same is for running. Right after you finish a race you don't say "I run," you say "I ran." Furthermore, even when your are running you don't say "I run," you say "I am running." I run only works as an answer to some question like "What do you do when..." or to say you run regularly.
The same works for the -to be- verb. You don't say "I am the president" right after your term runs out. You only say "I am the president" during your term.