Wiring 240v where 12/2 wire using a 20 amp circuit breaker is present
You would need to change out all the receptacles from NEMA 5-20 or 5-15 (125V) to NEMA 6-20 or 6-15 (250V) - it is a gigantic violation and hazard to supply 240V on a 120V outlet.

As such, running a new dedicated wire for the heater (if even needed) would appear to make the most sense, as you probably don't have many things with 240V plugs to plug in.