"I never asked to be born" is what fallacy?

It's a "Red Herring" logical fallacy. A child could use this argument to divert responsibility from himself back to the parent. For example, a parent telling the child he needs to be more responsible since he is 18 and if not he would need to leave the home. The child responds with "I didn't ask to be born".  Rather than arguing the point of being responsible he diverts it to "I didn't ask to be born". A Red Herring can be true, the child did not ask to be born. But he is trying to redirect the conversation from the original topic since being born has nothing to do with what was said.

First and foremost, statements like this can be signs of depression and a cry for help, so if this is someone close to you who said this, you should take it seriously.

As far as fallacies, this is just a statement rather than an argument, so you should ask them to expand on what they mean by this, then we can look for fallacies. For example,

"I never asked to be born. Therefore, my parents had no right to give me birth." This would be the Rights To Ought Fallacy , even though they would be legally (and arguably morally) wrong about the right.

Or we can say that this logic poses a paradox of sorts that is at least absurd (argumentum ad absurdum):

Only people to ask to be born should be born.
It is impossible to ask to be born prior to being born.
Therefore, nobody will ever be born.

This demonstrates their claim is absurd (assuming their position is "only people to ask to be born should be born.")