Can I convert a 120v circuit to 240v without running new cable?
IF the outlet is the ONLY outlet on the circuit, it's perfectly fine to change it to a 240V outlet (or disconnect, given it's for a heat pump) and change the breaker to a two-pole 240V 15A breaker - all your 120V wiring is already rated for 250V if not 600V. Since 15A/14Ga is the minimum wire size for house wiring, the wire size is already adequate by default for a 15A circuit.
If there are other outlets on the circuit, then there's going to be a problem, or more changes / rewiring will be required.
The white wire should be remarked red (preferable, IMHO, more obviously not just tape for tape's sake) or black (or any other &quot;hot color&quot; so not gray, white, or green) at both ends.