No, the Impossible Burger does not contain tree nuts or peanuts. The Impossible Burger does contain refined coconut oil. The FDA classifies coconut as a tree nut for food labelling purposes, but refined coconut oil is not considered an allergen.
Please exercise the same level of caution you would with any food. If you have a known allergy, please check with the restaurant about their use of that ingredient and cross-contamination risks.