I'm replacing a 240 baseboard heater with a 240 1500 watt baseboard heater. The original one was wired with 10-3 600 wire on a 30 amp circuit. Can I replace the 10-3 600 wire with 12-3 300 wire?

Clarify for us do you mean, 10-3 600 is 10 gauge, 3-conductor, 600 volts?
And 12-3 300 is 12 gauge, 3- conductor, 300 volts?

If so, yes...
(as long as the manufacturers installation instructions allow it also)

Why replace the wire at all? You are replace in a 240V heater with a 240V heater. Right?

Just leave it as installed, never hurts to have a bigger wire. Not sure about the
" 300" wire, all of mine is 600Volts.

Maybe I am missing something. Sounds like kshoag's plan is to put 12-3 on a 30 amp breaker?