Can a filling cause the tooth to become sensitive to cold and hot foods/drinks?

Yes it is possible: If the tooth had a deep "cavity" that needed to be filled, the filling may be very close to the nerve. This may cause some sensitivity. It is also possible that your filling is a little "high" and needs to be adjusted. On and off sensitivity is always possible with a new filling. If you do not have decrease in symptoms in a week or two or it gets worse, contact your dentist for further evaluation.

Yes: 1. Filling is not right: if the filling has void inside or ill-fitting margins, the nerve of the tooth will be irritated by bacterial coming from the leakage. The classic sign of nerve irritation is the sensitivity to temperature change 2. Filling is right: cavity may be deep and the stress from instrumentation or material shrinkage cause the temporary or permenant change to the nerve.