I wouldn't go smaller than 1.25" (32mm) on an mtb, not because of vibration, but because of handling.
The front geometry (trail, etc) has been worked out for 1.5-2.5" tyres. You drop below 1", and it will not be good.
I had a lovely 26" bike, built for 2" tyres. I fitted 1.2" slicks (continental sport contacts; they claim to be 1.3" but measure skinnier) for speed. Sure, it went fast. However, at speeds over 30mph shimmy kicked in. Switched to 1.8" tyres - no shimmy. I found that anything below 1.5" seriously affected handling at speed.