Pfizer Inc. said two federal advisory panels have recommended removing the most serious warnings on its Chantix smoking cessation pill, clearing a major hurdle in the pharmaceutical giant’s push to lessen concerns about using the medicine.

The move comes out of a joint meeting of the U.S. Food and Drug Administration’s Psychopharmacologic Drugs Advisory Committee and Drug Safety Risk Management Advisory Committee. The panels have recommended to the FDA that a so-called Black Box warning—noting the risk of psychiatric side...