My illustrator designed the cover of my book, but my publisher did not want to use his cover. They wanted their illustration department to do it. What I didn't know was that they copyrighted the cover and the interior of the book. Does this mean I can't use that cover or my illustration ever again?
It's their cover and their interior. Naturally they would copyright those.
If the rights to your book ever revert to you, you can publish it with whatever cover and interior you can get the rights to.
Normally you are the automatic copyright holder of original content, and assign publication rights (territory, language and media-specific) to a publisher or publishers for an agreed period. The publisher's copyright would extend to the layout and design, I would have thought, unless the contract has been drawn up to assign copyright entirely to them. Check your contract terms.