A photo or two would be helpful. Also, what type of stain was used.

 

I haven't had much luck removing any kind of stain using chemicals such as mineral spirits. Knowing the type of stain will be useful. But, since many stains usually penetrate into the wood (not much BTW, but can vary depending on how long it was left on before wiping it off) you may have to sand it out.

 

Unfortunately, I think you are in a difficult situation. You can not remove the stain by much. Given, it is a veneer, you need to be very careful to not damage it. You might do a wipe with mineral spirits and it may even it out a bit. You may just need to stain a bit darker.

We really need a picture and type of stain to help you.

 

I wouldn't be surprised if mineral sprits sent the stain deeper into the veneer.
I vote for a chemical stripper or paint it.

Good Luck

 

Non-eco paint stripper and a scrub brush

 

I would help a lot to have pictures. How are the results "wrong"; to light, too dark, wrong color, or uneven? What type of stain was used (oil base, water base, or dye) and the color you were trying to get? Also, how did you apply the stain to get the poor results. White oak is usually easy to stain but sometimes needs multiple applications because of the low absorption rate. Also most woods need careful fine sanding before staining to get even results.

White oak does not absorb stain as readily as some other wood so after attempting to remove some of it with a solvent, based on the original type solvent used in the stain, and then "very" light sanding may be the best you can do. You might be able to carefully use a cabinet scraper with a bit more control than sand paper.
You can then try re-staining. Read the stripper instructions, some need rinsing which might be difficult on panels.

If you try a paint stripper and a "scrub" brush make sure the brush has soft bristles that don't' scratch the wood.

Remember if you are unfamiliar with how a stain will go on your project do some test pieces first. Saves a lot of grief.