dye over oil stain?

What would be the effect of putting a dye over an oil-based stain (rather than the other way around)? Anything?

The dye would likely be much more ineffective, depending in part on the nature of the stain and how effectively it sealed the wood. Dye is meant to penetrate and color bare wood, if sealed very much it does so unvenly. Occasionally that will give a desirable effect, but that's a very rare case calling for lots of testing before using on an actual project.

I agree with Steve. Oil based stains contain a resin (varnish) that seals the wood surface as it dries. Once sealed, the wood will not absorb the dye. Dye requires absorption to color the wood.

I have a similar situation having stained a piece of cherry that has not produced the desired color depth. If the project were sanded would an alcohol based dye work?

Unfortunately, sanding might not be the best way to open up the wood to let it accept dye evenly. You would be better off removing the stain with a chemical stripper. (This is the reason to test the finishing system before doing any finishing on the actual project.