Did I damage my belly button? Is it permanent?

Thanks for your question.  Tanning a pink scar should be avoided.  A pink scar if tanned can stay dark permanently.  All you can do at this point is to avoid additional exposure.  

Without photos, it's hard to even begin to answer your questions.  In general, skin that has been stretched acutely will tan/burn more easily.  That's why we recommend tummy tucks, breast augs, facelifts, etc be careful not to get too much sun exposure in the first 6-12 months after surgery.  If you have hyperpigmentation, it might not end up being permanent.  Show it to your PS.  He or she may put you on a bleaching agent called hydroquinone.