Is there a Tantrayana (Tibetan Buddhism) training related to "burning" past bad karma?

Purification of karma is the same as purifying the mind. The mind holds all of the negative karma. When one attains nirvana, one has exhausted their negative karma. They become immune to the material reaction. So mantra, meditation, tantra, all of these have that aim. To purify the mind. It happens at many levels, conscious and unconscious. You can think of all the practices of meditation and yoga "burning karma." It sounds magical because of the burning word, but really it just is referring to ways you can rapidly purify and still the mind. This can be a very difficult process if it is made to be done expediently. For example, if you have done some black magic on someone in a past life you may have to experience some of the reaction. The purification process usually relies on the energy of a deity or a guru to assist in this process, to make it as manageable as possible. But one cannot think of this as any ordinary thing, one may have millions of years of karma to deal with. Most Buddhist practices are designed to deal with the negative karma. Ending bad thoughts, habits, through the puja, mantra, meditation...don't think of it as something extra special. Ending all your negative karma, ending all karma, is the goal.

In my opinion, you cannot change the past. We are now is the result of the past actions whether mentally or physically. There are some people who are ill and need to replace their failure organ with someone else's. However, not all organ replacement works well with the 'new' body.

But we can take care of the present moment and do good deeds as much as we can. This will add more 'water' so that the 'saltiness' of a glass of water can be reduced.

In the Theravada pali cannon:

In this Devadaha Sutta , the Buddha refutes the theories of the Jains — here called the Niganthas — and heaps ridicule on the idea of trying to burn kamma by asceticism. First he notes that none of the Niganthas have ever come to the end of pain by trying to burn it away in this way;

"Going to Niganthas who teach in this way, I have asked them, 'Is it true, friend Niganthas, that you teach in this way, that you have this view: "Whatever a person experiences — pleasure, pain, or neither pleasure nor pain — all is caused by what was done in the past. Thus, with the destruction of old actions through asceticism, and with the non-doing of new actions, there will be no flow into the future. With no flow into the future, there is the ending of action. With the ending of action, the ending of stress. With the ending of stress, the ending of feeling. With the ending of feeling, all suffering & stress will be exhausted"?'

"Having been asked this by me, the Niganthas admitted it, 'Yes.'

"So I said to them, 'But friends, do you know that you existed in the past, and that you did not not exist?' (*basically, have you seen your own past lives?)

"'No, friend.'

"'And do you know that you did evil actions in the past, and that you did not not do them?'

"'No, friend.'

"'And do you know that you did such-and-such evil actions in the past?'

"'No, friend.'

"'And do you know that so-and-so much stress has been exhausted, or that so-and-so much stress remains to be exhausted, or that with the exhaustion of so-and-so much stress all stress will be exhausted?'

"'No, friend.'

"'But do you know what is the abandoning of unskillful mental qualities and the attainment of skillful mental qualities in the here-&-now?'

"'No, friend.'

"'So, friends, it seems that you don't know that you existed in the past, and that you did not not exist... you don't know what is the abandoning of unskillful mental qualities and the attainment of skillful mental qualities in the here-&-now. That being the case, it is not proper for you to assert that, "Whatever a person experiences — pleasure, pain, or neither pleasure nor pain — all is caused by what was done in the past. Thus, with the destruction of old actions through asceticism, and with the non-doing of new actions, there will be no flow into the future. With no flow into the future, there is the ending of action. With the ending of action, the ending of stress. With the ending of stress, the ending of feeling. With the ending of feeling, all suffering & stress will be exhausted."