Most of what goes wrong here is not the shampoo being harsh. It is a product meant for one species ending up on another, and the one that matters is a dog flea product on a cat.
Plenty of dog flea shampoos, sprays and spot-ons contain permethrin. On an EPA approved label for one of them, a topical treatment that is 45 percent permethrin, the warning is printed in these words.
The reason is not that cats are delicate in general. International Cat Care puts it plainly: a cat's liver cannot break the chemical down into a harmless form, so it builds up instead. The signs they list are tremors, twitching, being oversensitive to touch and sound, walking unsteadily, and in bad cases seizures, trouble breathing and blindness.
If a permethrin product has gone on your cat, or your cat has been lying against a dog that was treated with one, call a vet straight away. Do not wait to see whether anything happens.
The two sources do not agree on how long to keep them apart, so here are both. The EPA label above says to keep cats away from treated dogs for 24 hours. International Cat Care advises 72 hours. If you have a cat in the house, take the longer number.
Natural is doing a lot of work on some labels. The ASPCA says that in their concentrated form, at 100 percent, essential oils can absolutely be a danger to pets, and singles out tea tree oil by name.
Three ways it reaches them, all from the same page. Oil placed on the coat or walked through. Oil swallowed, which for a cat means anything on the fur, because it is going to lick it off. And diffusers, which the ASPCA treats as the least of the three as long as the animal cannot get to the machine.
Buy a wash sold for the species standing in front of you, and read the active ingredients rather than the front of the bottle. If it lists permethrin and you own a cat, it does not come in the house. If your animal has a skin condition, the wash is part of treating it and your vet should be the one choosing it.
How often is the question everybody asks, and none of the sources above put a number on it, so this page will not invent one. It depends on the coat, the skin and what they have rolled in.