The title question is rather illustrative. I suppose the real question would be:
Is heat cumulative?
Put back into an example:
If I have a lit candle right beneath an iron bar, assuming the candle will remain lit indefinitely, and that the heat-losing rate is below the heat-getting rate. Will the bar eventually reach the needed temperature for it to melt?
If the answer is no:
Once the iron bar reached the max temp the candle can get it to. Where does all of the energy (heat) go after?
EDIT:
The question "is heat cumulative?" can be ignored as it is out of place and is misleading. Althrough the answer for it is "yes", it doesn't mean the answer for the general question is also "yes".
The point is not if it is actually possible to melt iron with a candle. Iron and candle are mere parts of the illustration, their properties are irrelevant.
A better phrasing of the main question would be:
Could a heating object contained in a perfectly closed system push the temperature of the system above its own temperature?
Answer
I'll try a simple explanation. Assume that there are no phase transitions initially. As you heat a body, its temperature rises, and it radiates energy into the surrounding space according to $$P = A\varepsilon \sigma T^4$$ ($\sigma$ is the Stefan Boltzman constant, $A$ is the surface area, $T$ is the temperature, $\varepsilon$ is the emmissivity).
Obviously, $P$ increases as $T$ increases. So, if before the bar reaches its melting point, $P$ becomes equal to the power input (from the flame), there will be no net flow of energy across the surrounding-bar interface (for if there were, $P$ would be greater than the input power, and more energy will be lost than gained, taking it back to the stable equilibrium point). At this point, notice that the temperature is constant, because any energy supplied by the candle is equivalently emmitted by the body (sort of dynamic equilibrium). Of course, this is only possible if the bar doesn't melt before this temperature is reached.
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