Will tea tree oil help with bed bugs in a mattress?

Probably not. If you have bed bugs, you need a professional pest control operator to treat your home. Put your mattress and box springs each in a sealed (zippered) cover. Seal the zipper with tape. Launder all clothes, linens, and bed clothes, and keep them in sealed bags. And call that pest control company, and make sure they know how to kill bed bugs (some don't!)