First of all, there are many Lunar Meteorites that have been found, but I'm assuming you don't mean those.
The first bits of the Apollo moon rocks to be touched were dust inadvertently touched by the 2 astronauts that landed on the Moon. All of them touched at least some dust samples on the return trip that were contained outside of the container. Some of it might have been deliberate, see the transcripts:
05 11 13 55 CMP How big are the rocks that you just scurried around and picked up with the tongs? Good gravy! Beautiful! Just crack those guys open and get a - you know, virgin interior of them in a vacuum, and they'll have a ball. ...
05 11 14 19 CMP Hey, the Velcro - the Velcro around them sort of ...
In addition, the workers who helped the astronauts get out of the Apollo capsule are known to have touched at least some dust. One of their jobs was to get dust off their suits by using tape.
The next candidate are the samples that were sent to world leaders from Apollo 11. While there is no record of the dust that was put in to the display being touched, it seems quite likely that it was at some point in time.
I have searched for some kind of a record of a scientist mishandling one of the moon rocks, and I can't find any definitive record of such.
There are 3 rocks that the public may touch with their bare hands. The first of these went on display in 1976.
In addition, there is a few Apollo rocks that have been stole, and likely touched.