The tripping breaker won't cause the heat source to be stuck 'on', so there's no worry of an unregulated heat source.
I'd recommend troubleshooting to see if the problem is in the breaker (relatively unlikely) or in the thermostat or some other device that is connected to the circuit before calling an electrician.
It could be that the circuit is overloaded (likely cause). Simply adding up the draw of each device on the circuit, and then comparing that to the amp rating of the breaker, is one way to do this. Another would be to unplug something that draws a good amount of power and seeing if that stops the tripping.
Plugging the thermostat into a different circuit would show if the problem is a fault in the thermostat (if it was, then this other circuit's breaker would trip). If that's the case, then replace the thermostat.
In any event, replacing a breaker can be done without paying an electrician. It is only slightly more complicated than screwing in a light bulb.
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