According to the CDC, "cloth face coverings should not be placed on young children under age 2, anyone who has trouble breathing, or is unconscious, incapacitated, or otherwise unable to remove the mask without assistance."
Beyond that, every state has its own guidelines regarding the wearing of face coverings. In California, the state's Department of Public Health gives mask exemptions to “persons with a medical condition, mental health condition, or disability that prevents wearing a face covering." This includes “persons with a medical condition for whom wearing a face covering could obstruct breathing or who are unconscious, incapacitated, or otherwise unable to remove a face covering without assistance.” In New York City, you don’t have to wear a face covering if you have “a health issue that makes you unable to tolerate [one].”
But here's where it gets tricky: None of the health agencies specify any health conditions that would constitute a mask exemption—and some health professionals don't believe these stated, unspecific exemptions for surgical masks or cloth face coverings are necessary. "There are no known medical conditions aside from a severe skin condition [like a very severe burn that needs medical attention] on your face that would prevent a person from wearing this type of mask,” David Kaufman, MD, pulmonologist and director of the medical ICU at Tisch Hospital, tells Health. “If you can wear a scarf to keep your face warm in the winter, you can wear a mask to prevent the spread of disease.”