Firewood Banks Act of 2022 This bill directs the Department of Energy, in collaboration with the Department of the Interior and the Forest Service, to establish a pilot program to provide heat energy for residences of low-income and disabled individuals with wood-burning stoves. Under the program, trees on federal land shall be made available to firewood banks and federal land may be made available for firewood banks. The bill also directs the Forest Service, using certain funds made available by the Infrastructure Investment and Jobs Act, to provide financial assistance for operating firewood banks. Cooperatives, nonprofit organizations, and state, local, or tribal governments may apply for financial assistance.