In spoken English, "I forget" is most commonly used as a very informal way of saying "I forgot". This is very common in casual conversation:
A: Do we have homework this week?
B: I don't know, I forget what the teacher said.
You could use "I forget (x)" to indicate that you always or habitually forget something, but then you would want to use the present tense in the inner clause, and probably use some modifier, like always:
John is such an unmemorable speaker. I always forget what he says.
"I forget what he said" could mean that you habitually forget one thing that he said at a particular time in the past: e.g.
I always forget what John said on that one August night. He gave me some advice, but I can never remember it.