House : House's leg

In episode "Three Stories," after they do the two surgeries to remove the blood clot and dead muscle tissue, had House let them finally amputate his leg, would his pain have ended? I'm not talking about phantom pain, though there is that. Just curious if anyone has any ideas.

Maybe not gone completely but would have been less pain i guess.

That pain might have gone BUT amputees are in all sorts of chronic pain. Not "just" the phantom pain. The more pain yhe person was in before an amputation, the bigger the probability of bad phantom pain. Wearing a prosthetic is uncomfortable, that skin isn't made to carry your weight. You need crutches when you want to pee, because you can't wear a prosthetic 24/7. As you grow older, the whole body gets weary, shoulders, hips, the knee of the former healthy leg suffers amputation is NOT a valid pain treatment.

If he had it amputated and was restrained to a wheelchair for the rest of his life? The chronic pain would have gone away, yes. But wearing a prosthetic all the time especially for someone that walks around all day like House would produce a different sort of discomfort.

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The chronic pain would have gone away

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No, the chronic pain wouldn't have gone away. Chronic pain develops in the pain center in the brain and in some cases even in the spinal cord or in interaction of the brain with the spinal cord. Once the brain learns the pain, it most likely will never go away.
One reason why House is in chronic pain is because the nerve endings, which have been damaged by the non-oxygenation during the infarction or have been cut off during the middle-ground debridement surgery (which was recommended by Cuddy), are still firring pain signals. If you amputate the leg, you will cause even more cut off nerve endings, which also will send out pain signals.

House to Wilson about Cuddy: She is not some floozy in a bar. She is the floozy I work for.

If they could amputate and have the Ketamine coma again to reboot his brain it might work.