In the Meat Locker, if you don't choose either of them...
 

What happens when you run out of time? Is it just a 'game over screen', also what happens if you don't choose a dialogue option quick enough?

If you don't choose a dialoge option, Lee is silent and people react to that as if you picked the "..."(silent) option.

As for if you pick no one, I read Kenny just says you are worthless, pushes you aside, and kills Larry. Then it acts more like if you try to save him with you against Kenny and not Lilly.

Oh right okay, thank you.