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If the resistor is the first thing to fail, I would guess it leaves an open. But that's just a guess. If you want it to leave an open, you should buy a special type of resistor, that is promised to leave an open in case of burn. Capacitors are also sold in the same manner.
I suspect the result would be a burned mess with some resistance - i.e. neither open nor short. I would worry about the fire hazard. Are you using a flame-proof PCB for this circuit?
You should also consider using some current limiting. Without knowing anything about you circuit, you could consider adding a series PTC thermistor for current limiting. The thermistor has a very low initial resistance, but when current increases, the thermistor heats up, building up more resistance, thus limiting the current. This method won't have a fast protection, nor does it cut current spikes, but it's pretty harmless and easy.
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