Learn Finnish: Vaihdatan renkaat autoon. - I am having the tires changed on the car.

It comes from the verb vaihtaa (to change). By adding the curative suffix -ttaa, it becomes vaihdattaa, which means "to have something changed by someone else". Notice the consonant gradation: the strong -ht- in the infinitive weakens to -hd- in the first person singular form vaihdatan (I have changed).

Why is the object renkaat in the nominative plural rather than the partitive?

Renkaat is the total object here. Because having your tires changed is a completed, finite action applied to the whole set of tires, the object takes the standard nominative plural form. If you were having an unspecified number of tires changed, you would use the partitive plural renkaita.

Why does the sentence use the illative case autoon (into the car) instead of a location case like "on the car"?

In Finnish, verbs related to changing, adding, installing, or attaching parts to a whole generally require the illative case (into). Conceptually, you are installing the new tires "into" the car as a complete working vehicle, rather than just placing them physically on top of it.