Good question DigIndig. It was all trial and error. I've done a lot of woodworking, but I've really never done a project which required regular finish AND paint. My first thought was to paint first, then sand off the paint in the areas I didn't want it and then apply the veneer. This turned out to be the worst option for me. The first reason was that sanding latex paint is ridiculous. All it does is gum up the sandpaper, and what I thought would be a very quick sanding job ended up taking much longer (and used a lot more sandpaper) than I had anticipated. The 2nd reason was that all that sanding along with trimming veneer caused a bunch of sawdust to get embedded into the paint. I ended up having to clean up the paint as best I could and spraying AGAIN after veneer anyway. The problem really was the sticky nature of latex paint. Even days after painting it was sticky enough to attract dust. If I were to do this project over again I'd probably try an oil based paint and see if it worked better.
Next I tried veneering and finishing the veneer before masking off the veneer and painting. My idea was that the finish on the veneer would prevent any paint from being absorbed into the veneer. The problem here was that the masking tape actually damaged the finish. I even used the yellow "frog tape" which is specifically designed for delicate surfaces. It actually ripped out a very fine layer of veneer. I had to sand the veneer all over again and then reapply finish to the veneer. The last option, which I thought would be the worst - was to veneer, then paint, then finish the veneer. It turned out that the veneer did not soak up paint in the least. That could be because latex paint is very viscous - different paint may behave differently. I also didn't bother sanding the veneer before paint, as I knew I'd have to sand again afterwards anyway. I still got a little bit of dust in the latex while sanding the veneer, but it was manageable.
All in all, that order (veneer, then paint, then finish the veneer) turned out to work the best for me. If I were to try again with an oil based paint you may find a different order works best. Also, if you finish your veneer with a film finish (conversion varnish, lacquer, polyurethane, etc.) then you may not have the issue with the tape damaging the finish when masking - but it sounds like you plan to use an oil finish as I did.