Keenan was born in Cambridge, Massachusetts. His family is Irish American and Roman Catholic. He has a twin brother, John, and two other siblings. Keenan attended Boston College High School and Columbia College.

In 1991 Cheers creators James Burrows and Glen and Les Charles, having read Keenan's novel Blue Heaven, invited Keenan to create a new sitcom for their production company. The resulting pilot, Gloria Vane, starring JoBeth Williams, was not picked up by a network, but it led to a writing post on Frasier. In 1992, his first play, The Times, a musical that charts the course of a seventeen-year marriage between Liz, an actress, and Ted, a writer, won the Richard Rodgers Awards for Musical Theater, awarded by The American Academy of Arts and Letters. In 1993, the lyrics for The Times won the Edward Kleban Award.

He joined the staff of the sitcom Frasier as an executive story editor in 1994 for the series' second year. His first produced script for the series, "The Matchmaker," received an Emmy Award nomination, a GLAAD Media Award, and the 1995 Writers Guild Award for Episodic Comedy. He won a writing Emmy Award in 1996 for being one of eight writers of the classic Season 3 episode, "Moon Dance," and also received Emmy Award nominations for "The Ski Lodge" episode in 1998 and, with Christopher Lloyd, "Something Borrowed, Something Blue," in 2000, which won the 2001 WGA award for Episodic Comedy. During his six-season tenure on Frasier he rose through the ranks from executive story editor to co-producer, supervising producer, co-executive producer, and finally, executive producer. He was executive producer when the series ended in 2004. He also wrote the series finale, "Goodnight, Seattle." Keenan won five Emmy Awards during his tenure on the show. He was nominated for Outstanding Writing in a Comedy Series five times, and won once. He won the Outstanding Comedy Series award four times for his work as the show's producer. He also won two Writers Guild of America Awards for his work on the series.

In 2006, Keenan joined Desperate Housewives as a writer and executive producer for the third season of the television show. Although his work received good critical response, including having one of his episodes,"Bang", named the best of the season by many critics, he chose to leave the series after one year. However he is still credited as being a consulting producer and even wrote the Season 4 episode Distant Past.

Keenan also created two short-lived comedy series with fellow Frasier producer/writer Christopher Lloyd: Bram and Alice in 2002 and Out of Practice in 2005. He also co-wrote the 1994 film Sleep with Me as well as the screenplay for the 2007 Annie Award-winning animated feature Flushed Away.

Keenan is also a published author, and is commonly referred to as a "gay P. G. Wodehouse". As of 2007, he has written three novels:

• Blue Heaven (1988),
• Putting On the Ritz (1991), and
• My Lucky Star (2006).

Putting On the Ritz won the Lambda Literary Award for Humor in 1991, and My Lucky Star won the Lambda Literary Award for Humor in 2006. In October 2007, the novel also won the Thurber Prize for American Humor.

Keenan lives in Los Angeles, and does not drive a car. Since 1982, he has been partnered with Gerry Bernardi.