Hi irinaofr,
"May I ask you not TO smoke?"
"I believe TO is needed here, right?"
Yes, I believe Matthew is correct. However, to me, in BrE this is not as polite as it may at first seem on the surface. It actually means that you are directly telling, or instructing, them not to smoke, but trying to make it sound polite by using "May I....". It would mainly be used by authority figures. It sounds rather old fashioned and most people now, even in authority, would probably use the forms recommended by Skrej and emsr2d2.