In the Star Trek: The Voyage Home, Chekov is captured after sneaking on board the USS Enterprise. During his interrogation we see three items of 23rd century origin:
- Chekov's Starfleet ID
I suspect that the phaser and communicator were actually of Klingon origin and came from the Bounty.
During this interrogation Chekov unsucessfully attempts to use the phaser during his escape and ends up just running away before falling and injuring himself. Kirk et al rescue Chekov from the hospital where he was taken and the movie goes on.
But what happened to those three 23rd century items? Even if the phaser was inoperable, it and the communicator would provide a wealth of information for any military research center.
To me, the leaving behind of future tech is akin to the leftover bits and pieces of Arnie that Cyberdyne Systems heavily researched (and we all know how that turned out!)