Nah I can't see that ever happening. You can talk about the Doctor being upset by himself, even hating parts of himself, but they'll never go as far as suicide. It is a children's show in the end, and even if you make his suicide non-violent, it's the message that counts.
Self-sacrifice and being okay with regenerating is something else, but actually wanting to end his own life won't happen. Vincent & the Doctor was the closest Who will ever get to discussing suicide, and that broadcast ended with a message on screen listing suicide hotlines, and a voice over talking about how viewers who were upset with the contents of the show should contact any of those numbers. Which is already pretty insane when you think about it - I'm very (happily) surprised the BBC allowed that episode to air.
But the Doctor killing himself because he hates himself.out of guilt/whatever? Not going to happen.
Also consider that, even with the dramatic final moments, final episodes tend to be more of a celebration of that specific Doctor, and a tribute to the actor. Painting him as a character who sees no way out apart from suicide* won't fly, and I can't see them sending off an actor with his character thinking 'this one didn't work, maybe the next one will turn out better'.
* except, of course, if it's to save someone else, to sacrifice his own life for another. That works and has been done in the past. But not the 'I don't deserve to live' kind, where he ends it out of self-hatred. That's how you kill the show, and alienate/scar a ton of kids.
The Doctor is, in the end, a character that, above all, celebrates life.