Corrected entry: Starfleet has serious regulations on bringing future technology to inappropriate time periods (which is why the Bird of Prey remained cloaked during it's time in 1986). Chekov being fully aware and considerate of these regulations, considering he has been in Starfleet for about 20 years should have never tossed the Klingon phaser (23rd Century technology) at the navy officer (20th Century person) with full knowledge that he'll never be able to retrieve it. He is too smart for it to have been panic.
Correction: First off, the men interrogating him do not appear to be Navy officers but rather federal agents (FBI, CIA, etc.) The guy asking the questions has some idea of what the phaser is supposed to be, "Make nice, give us the ray gun" so he either deduced that or as is common in interrogations, they're asking the same questions over and over. The interrogators obviously don't believe a word of what Chekov is saying, hence why the one agent tells him, "Go ahead, stun me."
Correction: Maybe a bad error in judgment, but a character mistake, not a movie mistake.
wizard_of_gore