We break up 3 months ago... On christmas I sent him a message : merry christmas:) he texted me back : merry christmas:D... Should I text him for his birthday?

You shouldn't have "texted" him the Xmas message. However, if you want to send a birthday card by regular mail. That would be OK. Texting indicates that you want a response.

If you are still friendly with him, sure, wish him happy birthday.

If you are thinking you might want to get back together, and he is not seeing anyone, go ahead. However, if he is seeing someone else, leave him alone and move on.

As long as its just a birthday greeting, and not trying to keep the door open for romance, why not?

No, go no contact and move on. Past relationships are meant to stay in the past, there is no need whatsoever so text an ex.

I see no problem with it as long as that is all there is.
Only three months after a break up why are you wanting to communicate?

I wouldn't bother. If he doesn't text you back you'll be hurt.

Wishing someone a happy birthday via text is just slightly better than the obligatory wall post.

What's the point? Unless you want to get back together or you want to be friends? IF you don't want either, then what's the point?