When yet in the temporal sense means "up until now", the present tense with yet would be unidiomatic as an expression of something which has not taken place so far or has not been completed:
He does not pass his driving exam yet. unidiomatic
We would expect
He has not passed his driving exam yet.
But we can use a verb in the present tense with yet if the verb refers to a state
He does not have his drivers license yet.
The meaning of yet with states is "as of now".
We can use it with verbs in the present tense if they express ongoing action rather than action which completes (telicity). In that case the idea is that something is not ready to happen or begin, that now is too early or too soon for it, as it will happen in the future:
He does not go on stage yet. His part is in the next act.
He does not take his driving exam yet. He is too young.
Those same facts could be presented as things which have not taken place so far:
He has not gone on stage yet. His part is in the next act.
He has not taken his driving exam yet. He is too young.
Things can get complicated:
I have yet to see a pink elephant.