Constantly ingesting non-fatal amounts of poison is called Mithridatization, after Mithridates VI of Pontus who so feared being poisoned by his enemies that he constantly ingested poison to build up his tolerance to it. According to legend, it worked so well that when he was about to be captured by the Romans, he tried to poison himself (because being killed by the Romans would have been a much longer and much less pleasant way to go), but didn’t die because he had built up his immunity so much. He had to ask his bodyguard to kill him instead.
This actually does work, to varying degrees, depending on the substance involved. There was a preacher who was slowly being poisoned by his wife with arsenic. She expected the poison to slowly build up in his system and kill him, making it look like he suffered from a long term illness and finally died. After a while she got impatient, and gave him a dose that was something like three times the amount that would normally be fatal. However, because he had been ingesting arsenic for so long he had built up enough of a tolerance that instead of dying he only ended up in the hospital.
Folks who handle poisonous snakes also sometimes inject themselves with increasing doses of snake venom so that they build up an immunity. It’s a smart thing to do if you are in this line of work since an accidental snake bite could easily be fatal.
There are some poisons that just build up in the body and kill you though, and even if you ingest low levels of them you don’t really ever build up a good immunity to it. So it’s not a universal thing.
These folks don’t end up becoming poisonous themselves, though. They just end up with a high tolerance to substances that would be fatal to you and me.
The closest thing I can think of to someone becoming poisonous is the case of Gloria Ramirez, aka the “Toxic Lady”. On Feb. 19, 1994, she was brought into the Emergency Room in Riverside General Hospital in California, in respiratory and cardiac distress, and soon went into full cardiac arrest. One of the nurses drew blood (standard procedure for this type of case) and noted a foul odor coming from the patient’s blood, and passed out. The doctor took a sniff from the syringe that the nurse used, and also passed out. Four other members of the ER staff also passed out. To make a long story short, the patient died, and no one ever figured out why the ER staff all passed out. It seemed that Gloria had somehow become poisonous, but an autopsy failed to find any toxin that could explain what had happened.
It’s not quite the same as the Vish Kanya in the OP, but it is a case where someone (maybe?) became poisonous to others.