You could probably strip the exterior of the existing doors off, but who knows how much work that will take.
On our fridge, the finish panel is just that. You can get the fridge in different finishes, so the panel must come off. The hardware (hinges, handles) all bolt into a frame that is not integral to the finish panel.
If you stripped this off and tried to replace it with wood, the only area I see difficulty is the clearance on the doors when opening/closign. I think this would be a problem with either a side by side or French door fridge. There is not much space between the doors when they are closed, so you'd need a pretty thin veneer to make it work. Probably not so much a problem with over/under styles.