^You're right -- we saw Kirk, Kor, and the others try to attack each other and then pulling away in pain. Spock's line was, "Inexplicable, Captain. Extreme heat. Not only the weapons but the bodies as well."
That went too fast for me, so I didn't catch it.
I've often speculated (though certainly not concluded) that it was more of a widespread manipulation of mind and sensory perception than true heat.
That makes sense. If the Organians really heated the weapons to 350° as claimed, then it would have likely caused smoking and skin burns. Neither of these things actually happened. Or at least we didn't see anyone get treated for said burns.