Most wood veneer is in the ballpark, in terms of thickness, of heavy fine art papers. You would want to have a printer with a relatively flat paper path to be successful. I have a client that paints on hardwood plywood, and would be curious as to how well the process works.
As Bill mentions, a flatbed printer, of which most are solvent or ecosolvent based, would likely be your best bet. If you could find one with a white ink available, I would think you would have the best results, as you are only going to be able to darken the tone of the wood with most fine art inksets. I could see a blond wood working, as long as the absorption is not too bad. There is always Inkaid, as an option.
If I was going to experiment, I would look into a UV flatbed and precoated veneers like Braewood prefinished veneer. Braewood comes with a phenolic backing, like plastic laminates, and a polyurethane topcoat. It is already finished and ready to print on (assuming you do not have any resist problems with the ink). There would at least not be any absorption issues. You could topcoat the print with another layer of polyurethane, if the sheen differential was not pleasing. Keep us posted if you happen to pursue this.
