Can a serve touch the net in volleyball?

Yes. Under current rules (since 2000), a served ball that contacts the net and still crosses into the opponent's court is a legal serve and play continues.

Full Explanation

Per FIVB Rule 12.7.3, a served ball that contacts the net (including the top tape) and crosses over to the opponent's court is a legal serve -- commonly called a "let serve" or "net serve." This rule was changed by FIVB in 2000; prior to that, any net touch on the serve was an automatic fault. Under USAV, NFHS, and NCAA rules, the let serve is also legal. If the served ball contacts the net and fails to clear it, or crosses outside the antennas, it is a service fault (FIVB Rule 12.8.1) and the opponent scores a point.

Rulesets covered: FIVB/USAV/NFHS/NCAA  |  Rule Reference: FIVB Rule 12.7.3