Can you put your dog on your taxes?

If by "claim" you are referring to an exemption, the answer is no.

Since your animals are pets, you may be able to deduct related moving expenses (assuming you moved during the year).

If you have a service animal (eg seeing eye dog), some expenses may be deductible.

If you have a working animal (ie an animal that generates income), the animal's expenses may be deducted as business expenses. Please consult an tax professional for more information.