I don't think it would be THAT big of deal
(sorry Aaron)...I made our dinette & used an oak veneer to match our existing 40 year old cabinets. The result of the refacing would be a GREAT upgrade!
A couple of hurdles you'd have to overcome though:
1. I would suggest sanding the old plastic off (or use chemicals?) to assure a good adherence to the substrate...I wouldn't glue the new veneer directly to the old plastic stuff, it probably wouldn't stick well. You'd have to do the same to paint with good long-term results.
2. If you were wanting to put it on tambour doors, I would think it would be easier to just replace the tambour...too many small pieces.
3. Have you considered replacing only the doors and just resurfacing the cabinets...either with veneer or by painting?
There are several threads here talking about veneers and similar ideas, I would suggest a search using "veneer" as a keyword.
Good luck in whatever you decide to do ~ maybe someday I'll get to see the end results!
Shari