Not for our latest cat, he had a medical condition that they didn't know he was going to survive so they were looking for a long term foster for him. Now that he's fully grown the vets are happy that he's ok, you wouldn't know there was anything wrong with him. Then after 2 years they let us fully adopt him and didn't ask for a donation, but we regularly donate to that shelter anyway.
Our older cat which we got a few years earlier we did have to pay a donation for him.