The actual "oil filled heater" most likely would not cause a fire, but as with any electrical equipment, there is a "heater control circuit" and an "electrical supply cord" that our outside the oil filled reservoir.
Any electric item plugged into an outlet has the potential to short out, cause a fire and do significant damage. There is supposed to be an "operational circuit breaker" to protect the outlet from overload and "in theory" would shut off power in the event of any malfunction of the heater circuitry. However, this protection circuit does not always function as designed.
To answer your question: There is very little chance that an oil filled heater would cause a fire, but it is possible. I'd only use it where there are no combustible materials nearby, certainly nothing combustible contacting it and I'd be cautious of leaving it unattended while in operation. I wouldn't focus on a fire at any moment, but I wouldn't be completely without concern of the possibility.
Here is a link to one such fire:
http://www.keystonerv.org/forums/sho...ht=heater+fire
If you notice, this was not an "oil filled" heater, but the fire originated in the cord, which is a similar component to any oil filled heater. Caution is necessary, fear of the heater is not needed......