Is it possible to shave down my teeth to whiten them?

White spots on teeth are often superficial, meaning that with a little abrassive polishing of the teeth the stains can be greatly diminished or even completely eliminated. If the only cosmetic issue is white spots I would always recommend trying to eliminate the spots using microabrasion techniques before resorting to veneers.  

Shaving down your teeth is not the best option because you open yourself up to more issues down the road.  If the discoloration is not too deep, micro-abrasion can be done  to minimize or even remove the discoloration.  However, if it is deeper, the dentist may have to remove the stain/discoloration with a handpiece and replace it with bonding that  matches the surrounding tooth structure.  There is an option of prep-less veneers that can mask out the stains/discolorations and give your teeth a uniform appearance.   Best of luck to you.

It is not recommended to shave down discolored teeth without restoring them.  The reason being that nerve endings may become exposed and cause sensitivity.  It could also have esthetic and functional consequences depending on how extensive the discoloration is.  The discolored areas may be removed, but should be restored with either bonding or veneers.  Thank you, Dr. Elizabeth Jahanian.

Find a dentist in your area that uses ICON-A new conservative bonding agent that was made to help those with white spots.  You can try whitening strips to brighten your smile and then use the ICON.  Much success-Dr. Wendy spektor 

Your dentist is correct about veneers being the best approach to getting rid of the stains on your teeth.  Since the stains are intrinsic it is impossible to bleach them out.  If you shaved the teeth down enough to correct the problem you would be VERY unhappy because the teeth would be sensitive and painful frequently.  They would also be suseptible to decay .  I guess in a way that would work because you'd end up needing crowns which would solve the esthetic problem, but honestly, veneers are easier!

Removing tooth structure to remove stains is not the best approach, unless you were doing veneers though I understand your financial concerns you want to make sure you don't make matters worse by shaving teeth down you will be extremely sensitive & leave teeth more open to decay.

You may be a great candidate for Prepless Veneers (brand name Durathin, Lumineers). There is no shaving and no reduction of enamel. 

In some very FEW cases, stains are superficial, and simply roughening the outer layer will remove stains.  However, it is impossible to know how deep the stains are until you are "in there" and removing enamel.  You and your dentist need to be prepared with alternatives in case your situation can NOT be corrected easily.  

Porcelain veneers are the most predictable and likely solution to your issue.