No, you should not "just" replace the breaker with a 40 or 50 amp one.

Where are you seeing that the NEC requires 8/2 and 50A? It seems like 30 amps should be adequate for a 5000 watt heater - does DuraHeat specify a larger circuit?

1.25continuous load x 5000w / 240V = 26amps

The manufacturers instructions specify a minimum of 10 ga wire, which should be plenty for the 21 A load on a 30 A circuit, though it does specify SJO or SJTO. Your romex may not have a sufficient temperature rating (75 is specified, some NM is only 60 for the cable even though the individual wires are 90) to connect it directly to the heater. Or there might be a vibration issue that wants' a stranded wire. You may need to take the romex to a box and then SJO to the heater. Check the labeling of your wire and with your inspector.

I believe that typically the manufacturers specifications take precendence over the code prescriptions.