The Vedas, which are considered sacrosanct in all surviving sects of theistic Hinduism. There were once atheist sects like the Ajivikas and the Charvakas, but they went out of fashion a long time ago.
The verses in the Vedic texts can be classified in many different ways. For instance, “Bheda Stutis” and “Abheda Stutis” in the Upanishads (which essentially are Dualist and Monist verses). The classification that I’m interested in touching upon here is the distinction between Karma-kanda and the Jnana-kanda (which translate to procedural canon and the theoretical canon). The procedural part mostly deals with the details on rituals and worship (one bit that I vividly remember is the part on the rituals of animal sacrifice, which might come as a bit of a surprise). The theoretical part deals with the questions around the true nature of Brahman (the ultimate god) and Atman (the individual soul). Hindu priests usually specialise in the former (the part involving rituals) rather than the latter. I wouldn’t trust a lone Hindu priest’s opinion on such doctrinal matters. :)
The Vedic doctrine on your question is fairly unambiguous (which is quite rare for anything in the Vedas). The soul (aka the Atman) always takes precedence. The body is considered to be a mere shell in which the soul temporarily resides. Thinking that the body and the soul to be inextricably linked to the body is considered a common fallacy which even has a Sanskrit term for it (Dehatma bhranti). The body (Deha) is considered temporary and dispensable and the soul (Atman), eternal. What happens to the body after death is merely a procedural implementation detail and will not affect your chances at the lottery to attain Moksha. :)
The priest you spoke to probably is confounding the cremation rituals with the idea of sacrifice (aka Bali).
1. You could argue back to the priest that your organ donation is going to save lives which God would appreciate anyway and have your way :D
2. More broadly, one of the lessons of history [0] is that what begins as a practical way to do something becomes codified into religion because it's far more efficient to get the message across generations. I'd like to believe that burning the body (as many Hindus do) is one such [ Not burning the body may have caused infections and what not in tribal settlements long long ago ] . This being the case, organ donation is a safe way, too. If you're the more scientific kind, you can use this to further convince yourself to donate the body. I know I would use both 1&2 :)
[0] I read about this when researching how Halal and some other practises came about. I will try to dig up the reference but it was ages ago :(
Hinduism has no central doctrine, so while this particular priest may hold a view on organ donation (or lack thereof), there will be others who may encourage it. As an example - Lord Ganesha's head is "donated" from an elephant. The concept of organ reuse isn't new. There just wasn't the technology to deal with it in the past.
I may not be as learned as the priest you know. But I have yet to find any of the priests I know tell me organ donation goes against any scripture.
https://www.bbc.co.uk/religion/religions/hinduism/hinduethic...
https://www.organdonation.nhs.uk/helping-you-to-decide/your-...
(This is just from a quick Google search)
So here's a (silly) attempt to address the priest's argument directly: what if you lose a part of your body long before your death? Is that okay? If yes, returning an organ a couple decades later than planned shouldn't be that big a problem, no?
BTW, in my religion, saying "I am not that smart and don't know how to..." is a sin :p
Since the topic of organ donation came up, I thought of asking this question. This is a real incident. I am not making this up. I met a learned priest. Both I and the priest are hindus. He argued that since God gave us our body and organs at birth, we need to give it back to God the body and all the organs at death. I don't agree with the priests argument. It is just that I am not that smart and don't know how to counter that argument. I am sure there are hindu readers in HN community. Any thoughts on this?