My husband mashed his finger in the car door. Do we apply ice or heat?

Ice. In the acute phase of an injury like this, ice would generally be recommended. Depending on the extent of the injury, he may need to get it evaluated for a fracture. Hope he feels better soon.
Ice. First, look for any obvious deformity like fingers or other parts of the hand look bent or twisted out of shape...This is even more concerning if there are breaks in the skin where the deformities are. This would require getting seen soon. If that's all ok, then ice, (some sort of cold compress) to reduce pain and swelling, along with elevation of the hand will certainly help.