Fleischer was born in Washington, D.C. and attended Southampton College of Long Island University. As a child, he is reported to have spent several summers at Kamp Kewanee in La Plume, Pennsylvania. He is known as the voices of both Roger Rabbit and Benny The Cab in Disney and Amblin Entertainment's 1988 live action/animated hybrid film Who Framed Roger Rabbit. After that film's success, Fleischer continued to perform the voice of Roger in several television and theme park appearances, and in three follow-up short subjects to the original film. Other voice roles for Fleischer include The Polar Express and We're Back! A Dinosaur's Story.
On-screen roles include Back to the Future Part II and Gridlock'd. Fleischer has had roles on TV shows: he has had a recurring role on the 1970s hit TV series Welcome Back, Kotter as Carvelli, as Chuck on the hit ABC series Laverne & Shirley, and on the popular Disney cartoon series House of Mouse as the voice of Benny The Cab. He guest starred in 'The Weird Al Show as a guy in a band.