Generally not
A number of common fouls are listed in Quidditch Through the Ages, with descriptions. Many of them have a description of the kind of player who commit them in parentheses. For example:
Name: Haversacking
Applies to: Chasers only
Description: Hand still on Quaffle as it goes through goal hoop
(Quaffle must be thrown)
—Quidditch Through the Ages
If only Chasers can commit this foul, this would seem to indicate that each position has a specific role. Even more telling is the description for "Quaffle-Pocking":
Name: Quaffle-pocking
Applies to: Chasers only
Description: Tampering with Quaffle, e.g., puncturing it so that it
falls more quickly or zigzags
—Quidditch Through the Ages
If only Chasers can commit the foul of tampering with the Quaffle, this should be a strong sign that anyone else touching the Quaffle (except the Keeper) is already a foul. On the other hand, the Keeper certainly can legally touch the Quaffle, and is not mentioned in this foul, which is an inconsistency regardless of how one looks at things.
Similarly, there are fouls that can only be committed by Keepers, related to guarding the goal, or by Beaters, related to hitting Bludgers, indicating that these other positions also have specific duties.
Most telling of all is the foul of stooging:
Name: Stooging
Applies to: Chasers only
Description: More than one Chaser entering the scoring area
—Quidditch Through the Ages
Without a previously defined idea of who qualifies as a "Chaser," this rule would be meaningless (say, if Beaters could suddenly grab the Quaffle and switch to Chasers).
It is true that only the specific foul of a non-Seeker touching the Snitch is mentioned, but this is not surprising. The list is of common fouls, and certainly the heavy incentive for catching the Snitch could lead players such as Beaters or Chasers to abandon their positions. By contrast, a Seeker would be unlikely to leave the Snitch entirely for the other team in order to score a goal, and therefore such a foul would not bear mentioning.