IMHO....
Your pet most likely does not care what colour it is lol
Cats, well, most cats don't really like baths much all around and they groom themselves by licking all the time so I doubt I'd dye my cat - I do lionshave her tho! If I don't she hucks up huge hairballs all the time, even with brushing and I'm worried one will get stuck, so sure, I lionshave her - and I LIKE the way it looks. She doesn't go outside, she's never cold, she is happier without all of that hair actually."wild" cats do not have the ridiculous amounts of long hair that we have bred into some of our domestic cats.
If the dying can be done safely, with nontoxic dyes with no harm to the pet, sure, go for it, if it makes you happy your dog will probably be happy too - I coloured parts of Phoebe with kid-safe water-washable TOTALLY non-toxic crayola markers one Halloween and she LOVED The attention it got her, she got to sit out front at work with a sign saying "Please don't feed the dinosaur" and she was so happy, she loves people. My dog is really barely a dog at all, she certainly doesn't think she's a dog as no one's ever told her.
I think it's up to personal opinion.
(Dyed easter pets or injected/tattooed fish I don't agree with, those practices are not safe or humane, but I don't think that's what this topic was about)