Can a tumor produce something not currently found in our own bodies?
If you look at a cancer textbook (eg. Robert A. Weinberg's The Biology of Cancer), you'll find that by definition, a tumor arises from tissues normally found in the patient's body. There's a great deal more to it than just that simple statement, but given the context of the question, I think that answer should suffice.

In writing, "...tumors growing a tooth, hair...", I think you're talking about teratomas. These contain "normal derivatives of all three germ layers". These arise naturally from cells "...typically found in the human body..."

Cancer is a process where mutations in a single cell transform that cell into something that is by definition not "typically found in the human body." But a human cell transformed by mutation into a tumor (benign or malignant) is never (to the best of my knowledge) going to magically become a cell from another species or a very distant ancestor (ie. something that is not "typically found in the human body.").