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 1992, his first play, The Times, a musical about two gay men in an 18 year relationship, 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. Gradually gaining more influence on the show, he was executive producer when the series ended in 2004. He also wrote or co-wrote several episodes of the show, including 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 been 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). My Lucky Star won the Lambda Literary Award for humor in 2006. In October 2007, the novel won the prestigious Thurber Prize for American Humor.