Davis graduated from Hall High School in Little Rock in 1963. He graduated from the Arkansas Arts Center with a Bachelor of Fine Arts, followed by work with the Oregon Shakespeare Festival, the Stratford Shakespeare Festival, and six years with the American Conservatory Theatre (ACT). During his time at ACT, he taught acting classes.
On Broadway, he played Antonio Salieri in Amadeus in 1980, Maurice Montesor in Wrong Mountain in 2000, Oscar Wilde in The Invention of Love in 2001, Georges in La Cage aux Folles in 2004, George Bernard Shaw in The Frogs, also in 2004.
Off-Broadway, he played Rubin in Lake No Bottom in 1990, the Duke of Buckingham in The Tragedy of Richard III, Graham in the Alan Bennett monologue "A Chip in the Sugar" for the series Talking Heads in 2003, and he starred as Gaev in The Cherry Orchard in 2011.
Davis played his most famous character, Niles the Butler on the television series The Nanny, throughout its run from 1993 to 1999 and in its reunion special in 2004 (in cameo flashbacks only). Niles was known for his frequent use of deadpan one-liners, usually insulting character C.C. Babcock, whom Niles eventually fell in love with and married in the series' last season.
In 2000, Davis was nominated for a Best Actor Tony Award for his role in David Hirson's Broadway play Wrong Mountain. In 2003, he appeared in the Alan Bennett play Talking Heads. In 2004, he portrayed George Bernard Shaw in the Stephen Sondheim musical The Frogs. He co-starred in La Cage Aux Folles with Gary Beach from November 2004 to March 2005. (Reportedly, he clashed frequently with Beach and others. He was eventually replaced by Robert Goulet.)
In 2002, Davis guest-starred on the television series Frasier, playing Dr. Shafer in Season 10, Episode 8, "Rooms with a View".
In 2007, Davis was a member of the cast of an audio production of the play Blue/Orange, in which he took the part of Dr. Robert Smith.
In July 2008, Davis portrayed King Lear at the Shakespeare Theatre of New Jersey, under the direction of Bonnie J. Monte. That year he also guest-starred on the television series Ugly Betty.
In December 2010, he guested on The Fran Drescher Show through Skype.
In 2012, he played Prospero in The Tempest at Hartford Stage.
In 2016, he appeared on Broadway as Selsdon Mowbray in the Roundabout Theatre revival of Noises Off at the American Airlines Theatre.
