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I've started noticing people using constructions like "something won't do something" as a present tense instead of "something doesn't do something".

For example, here is a piece from Eminem's song

but he keeps on forgettin'
What he wrote down, the whole crowd goes so loud
He opens his mouth, but the words won't come out

My question is is it ok to say like this and is there any specific rule that says in which situations one can use won't when they would normally use don't/doesn't?

The example is using figurative language to describe the scene. Don't come out would be the most grammatical form to use in that situation, but the song is trying to evoke the sense of a struggle, a fight between the character and the words.

Don't is a more neutral term. It's a simple statement of fact - the words do not come. The line is about the singer's surprise that the words he's expecting don't come.

Won't personifies the words and grants them agency - they refuse to come out, while the character is trying to make them. The line is then about the singer fighting, trying to make the words come out.

There, won't means "refuse to". Similar contexts can be found among the results here.