During his early acting career, Drier also appeared in three ABC Afterschool Specials, in one of which, Hewitt's Just Different, Drier had a lead role as "Willie Arthur", the friend of the developmentally disabled title character. His late 1970s and early 1980s roles included When Every Day Was the Fourth of July (1978) and Peter Benchley's thriller Hunters of the Reef (1978). Other teen roles consist primarily of biographical dramas; most notably, Drier played a young Mickey Rooney in the 1978 Judy Garland biography Rainbow. 1978 also saw the filming of the made-for TV Jack Albertson vehicle Charlie and the Great Balloon Chase, which was not released until three years later, in 1981. In the 1980s made-for-TV movie, Homeward Bound, he played a terminally ill young man, Bobby Seaton, who spends a last summer vacation to repair his relationship with his father Jake, played by David Soul.

Young Drier's first role was as a deaf boy in two 1972 episodes of Lassie. He began his television career as a recurring performer on Rowan & Martin's Laugh-In from the middle of season three to the final season in 1973, hosting a "Kid News for Kids" segment. During this period, Drier had movie roles in the 1972 Jack Lemmon comedy, The War Between Men and Women, the 1972 Barbra Streisand comedy Up the Sandbox, and the made-for-TV comedies Roll, Freddy, Roll! (1974), and All Together Now (1975). In 1977 he was cast in Oh, God! starring John Denver as Jerry Landers and George Burns as God.

During the late 1990s, Drier accepted minor roles in sci-fi space-ship hijack thriller Velocity Trap (1997), and Storm Trackers (1999), a thriller about a secret military weather control machine gone awry. Since 2000, he has specialized in voice-over work in such films as Teaching Mrs. Tingle (1999), American Beauty (1999), What Lies Beneath (2000) Shrek (2001), 40 Days and 40 Nights (2002), The Shape of Things (2003), Jungle Book 2 (2003), the Lion King 1½ (2004), The Chronicles of Riddick (2004), Hauru no ugoku shiro (Eng: Howl's Moving Castle) in 2004, and Madagascar (2005).

Drier directed episodes of such series as Reba (2005) and Too Late with Adam Carolla (2005). He directed a well-received children's musical, Precious Piglet and Her Pals at the Whitefire Theatre in Sherman Oaks, as well as the critically acclaimed Love Like Blue in 2007, also at the Whitefire Theatre.

At the age of ten, Drier began voice acting as a regular character in ABC’s 1974 These Are the Days. Other recurring television roles included "Howie Borden" on The Bob Newhart Show and on CBS’s short-lived series Executive Suite as B.J. Koslo. He made appearances on The Waltons (1973), Adam-12 (1973), Apple's Way (1974), Police Story (3 episodes, 1974-75), Emergency! (2 episodes; 1975), Doc (1975), and Little House on the Prairie (1976), CHiPs (1980), Family Ties (1983), Kids Incorporated (1984), Diff'rent Strokes (1986), The A-Team (1986), Highway to Heaven (1986), Blacke's Magic (1986), Cagney & Lacey (1986), Hunter (1986), Just the Ten of Us (1988), The Munsters Today (1990), and Jack & Jill (2000).