What happens if I leave the head in?

Short answer - nothing much.

hypostome

Even when you remove the tick properly, its multi-barbed hypostome usually remains embedded in the skin. If you have carefully removed the tick's head and body, the hypostome itself is essentially harmless and will eventually fester out like a tiny splinter.

When there are only one or two tick bites, physicians sometimes dig out the hypostome using a big hypodermic needle or excise that section of skin with a biopsy punch.This may or may not be a good idea on people, but is certainly impractical on pets.