Can a pet scan tell if a tumor is malignant? I know that pet scans can find tumors, but can they tell if a tumor is malignant? Or does a biopsy always have to be done to tell that?

No, usually biopsy. Pet scans detect increased metabolism or consumption of some chemical, usually glucose. Many different diseases can use extra glucose - infection, healing after surgery, tumors (benign and malignant). If no exact diagnosis is known, the pet is positive and the pet scan and anatomy scans like ct do not look like something benign or easily treatable (infection), biopsy is usually performed.
Not definitively. Pet is an imaging technique that uses sugar that gives off a small amount of radiation. The sugar is taken up typically by organs and cells that are very metabolically active (using a lot of energy). Some examples include brain and liver in normal tissue but also occurs when infection or tumor is present. So biopsy is more definitive if tumor suspected but not always necessary. No test is 100%.
PET . Pet scans work by detecting metabolic activity in the body. Since one of the characteristics of malignancy is high cellular turnover and high metabolic activity, a mass that has high uptake on a pet scan is more likely to be malignant than benign. However, there are many things that can also produce high uptake on pet that are not malignant, including infection, inflammation, or even muscular activity. Conversely, there are also malignancies that don't produce high uptake on pet, because they are tumors of lower cellularity or less intense metabolism. Tumors smaller than 1 cm may not be detected by pet. A biopsy is still the best way to know for sure if a mass is malignant or benign, but pet is one of the best non-invasive tests to evaluate if certain masses are malignant or benign.