McDermott was born in Detroit, Michigan to parents who supported the arts and encouraged his love of reading. McDermott began studying art at the age of four when he started taking Saturday workshops at the Detroit Institute of Arts, the museum in his hometown. There was basic art instruction in the morning and then students were encouraged to sketch from the various collections of the museum.

In elementary and middle school he continued to sketch and paint, acted in a weekly radio program, and he studied ballet as well as music. All of these creative experiences would help McDermott create animated films and books later in life.

McDermott continued his education at Cass Technical, a public high school for the gifted with a notable art program. In 1959 he was awarded a National Scholastic Scholarship to the Pratt Institute of Design in New York City. As an extracurricular summer project McDermott decided to produce an animated film and chose The Stonecutter, a story he had loved as a child. McDermott conducted extensive research into the cultures and customs of the story's origins. During his junior year at Pratt in 1962, McDermott took a leave of absence to work as a graphic designer for WNET, a New York educational television channel. He also produced several animated pieces for The Electric Company on PBS. He received his Bachelor of Fine Arts degree from Pratt in 1964.

He produced three more short films soon after graduation Sunflight in 1966, Anansi the Spider in 1969, and The Magic Tree in 1970. In 1970 George Nicholson, a children's book editor, contacted McDermott and suggested adapting his films as picture books. The first was Anansi the Spider: a tale from the Ashanti, published by Holt, Rinehart, and Winston in 1972. It retells an Ashanti folktale explaining how Anansi was rescued by the separate contributions of his sons. Meanwhile it explains how the moon came to be in the sky. The Magic Tree, Arrow to the Sun, and Stone-cutter soon followed as picture books. Arrow to the Sun (1974) was developed simultaneously as a film released in 1973, McDermott's last animated film.

In the 1980s he published two books that paid homage to the traditions of his Irish heritage: Daniel O'Rourke and Tim O'Toole and the Wee Folk. During this time he also illustrated some books written by Marianna Mayer. In the early 1990s McDermott began working on a series of trickster tales and two works of mythological creation stories.

He died on December 26, 2012