I would suggest you consider religious attitudes to tattoos and body modification generally to inform a likely outcome, combined with beliefs about the value of animal life, and humanity's place within the animal kingdom. I'll speak about tattoos as a baseline because body modification is a similar theme.
Abrahamic faiths generally disapprove, the most obvious reason is this:
(Leviticus 19:28): "You shall not make gashes in your flesh for the
dead, or incise any marks on yourselves: I am the Lord."
This can be taken literally, and in the case of furryism probably metaphorically too. Depending on how the process works, "incise any marks on yourself" may be applicable. More generally this prohibition is consistent with a general theme of bodily purity.
Christians can focus more specifically on verses like Corinthians 18-20:
"18 Flee from sexual immorality. All other sins a person commits are
outside the body, but whoever sins sexually, sins against their own
body. 19 Do you not know that your bodies are temples of the Holy
Spirit, who is in you, whom you have received from God? You are not
your own; 20 you were bought at a price. Therefore honor God with your
bodies."
Here the point is that a person can sin against their own body, that their body is not their property, and that their body is a temple for honouring God. None of this is compatible with body modifications.
Judaism (especially), Christianity, and Islam are not fond of tattoos or body modification for similar reasons. Though in Shia Islam tattoos are not forbidden, strictly speaking, just discouraged. Shia Islam is also unusual in that sex reassignment surgery is state-sanctioned in the Islamic Republic of Iran, justified along a rather curious line that Allah permits man to process grain into bread; therefore transforming someone so their sexuality aligns with their gender is preferable to them being homosexual.
Generally the point is this: these religions consider body modification to be sinful because it rejects God's design. Furthermore, God created humans in his image, and only humans have immortal souls or the chance of passing into heaven. To choose to become a beast of sorts would be a spiritual degeneration, making yourself unclean.
In Islam the point is even more explicit: Allah made everything in the world perfectly. Consequently when people try to create artistic works from life this insults Allah, as no one could hope to match the perfection of creation - they would only ever produce a mockery of it; satirising and thus disrespecting the creator.
If you look to the art created by fundamentalist Muslims, it is almost always simple geometric patterns, calligraphy, and generally abstract things. Shia Islam is far less strict about this, and even has little problem with paintings of their prophet. But even then... there's no way body modification to the extent of furryism would be acceptable in Judaism, Christianity, or Islam. It would be a mortal, perhaps unforgivable, sin.
All religions tend to be obsessed with notions of purity and human supremacy, so even if we consider Buddhism and Hinduism to be more likely to be tolerant because they have a greater respect for life generally than Abrahamic religions, they still have a set of values which gives meaning to the form one's existence takes. In Buddhism and Hinduism reincarnation is a natural and purposeful cosmic balancing. So if one is human there's a reason for that. To change species may be seen as a perversion of that natural order.
Buddhism is not keen on beautification, as this is evidence of attachment; which is contrary to the aims of the religion. Buddhists want to realise non-existence, that their individual ego is an illusion, and to break the cycle of reincarnation which happens because of emotional attachment to the physical world which creates suffering.
Furryism in this context is sinful because it works to enforce the illusion of the self, who makes this decision selfishly for self expression or beautification. Whatever the reason, the outcome reinforces the things Buddhism seeks to escape: attachment to self, attachment to the physical, the creation of suffering by indulging in desire. So the fact Buddhists have a great deal of respect for all life, and would not want to hurt someone who did this, is actually besides the point.
Hinduism is less clear cut, but I suspect it's not going to be much different. Feel free to correct me, but given how Hindu fundamentalists tend to be violently intolerant of homosexuality, they will likely disapprove for a plethora of reasons already stated. But I don't know for sure.
Shinto may be the only religion which tolerates furryism. Japan's native syncretic religion is grounded in the existence of nature spirits, which must be respected. Tattooing, though recently something of a taboo for its associations with organised crime families, is not considered sinful. Combine this with the breadth of their spiritual perception, and it doesn't seem likely that Shinto would reject furryism in the same way or for the same reasons as the other religions mentioned. In the same way as tattoos in Japanese history sometimes have spiritual significance, furryism may be considered similarly; a form of spiritual self-expression.
Bottom line: With the possible exception of Shinto, this will be considered sinful. It violates the sanctity of creation, indulges selfishness, or just attempts to usurp the natural order.
