There are other therapies besides drugs and basic talk therapy, so at the very least I would suggest the person try all of those first if I was the doctor..
That noted, removing a limb that is useful (not cancerous, badly injured, super painful, etc.) seems unwise in general, and the person who doesn't want it can effectively destroy it him/herself if he/she wishes, by immersing it in freezing cold water for an extended period of time (I saw a program in which a guy did exactly this), and probably by other means as well. This would remove any moral dilemma from the surgeon, assuming he acted ethically in prior interactions with the person..
Given these circumstances - plus the potential for new treatments that might work to get the person to accept the limb as a positive - I would be hesitant to promote removing a healthy limb, and would certainly not want to pay for it to any extent through taxation or insurance costs, etc.
As for "being allowed", I suppose it might be OK if the amputee was paying for the operation and associated costs him/her-self and could be proven mentally competent to make that decision, though this would be a pretty high standard to meet I think given the decrease in functionality that would occur by way of getting a perfectly functional limb removed.
so desires? There's a condition in the brain that causes some people to want a limb removed, that can't be cured by drugs or talk therapy.