Born in Newcastle-under-Lyme, Staffordshire, Heath started his playing career at the Stoke City having joined the club from playing in the local Lads and Dads League. He signed a professional contract at 17 and played in the reserves under the guidance of player coach Howard Kendall, Heath's impressive performances helped the second sting to 2nd in the Central League and saw him win the clubs young player of the year award. Known as 'Inchy' due to his small stature he burst on to the first team scene in 1979–80 displaying great maturing in the First Division and scored two vital goals at the end of the season which ensured Stoke's survival. He scored seven goals in 41 matches in 1980–81 and five in 20 in 1981–82. With Heath fast becoming one of the most sought after midfielder in the county Richie Barker allowed him to join Everton in January 1982 for a then club record fee of £750,000.
He became Everton's top scorer in his first full season, scoring 18 in all competitions in the 1983–84 season. Despite the fact that Heath only played 17 games in the 1984–85 campaign, his scoring got into double figures every season until 1988. A goal for which many fans remember Heath is credited with changing Everton's fortunes in the 1983–84 road to the League Cup final at Oxford United. Grabbing a poor Kevin Brock back-pass, Heath scored a goal away at Oxford to equalize the match and earn Everton a chance to turn things around.
Heath then had a spell in Spain with Espanyol. After a single season he returned to England, where he spent the remainder of his career. In 1990 Manchester City manager Howard Kendall signed Heath from Aston Villa. Heath was one of a number of former Everton players signed by Kendall, and the transfer marked the third different club at which Heath had played under Kendall. His debut came as a substitute against Charlton Athletic in January 1990. During the 1990–91 season Heath forged a strike partnership with Niall Quinn, the short and agile Heath contrasting with Quinn's aerial ability. The pair started 33 matches together, though Heath scored only one goal, in the second match of the season. Heath then had a barren run that lasted 46 matches and came to an end in November 1991, when Heath scored two goals in a League Cup tie at Queen's Park Rangers. As the 1991–92 season progressed, Heath lost his place in the team to young striker Mike Sheron.
In March 1992 Heath briefly returned to his first club Stoke City, until the end of the 1991–92 season. In that brief spell back at the Victoria Ground Heath played in six league matches a play-off match against Stockport County and also played in the 1992 Football League Trophy Final which saw Stoke beat Stockport 1–0. Heath also had a notable three year stint at Burnley from 1992–95, where he helped them lift the 1994 Second Division play-Off trophy. Following a brief period at Sheffield United, he returned to Burnley as player-manager before retiring at the end of the 1996–97 season.
Heath was appointed manager of USL-1expansion teamAustin Aztex in February 2008. Austin Aztex had a close relationship with English side Stoke City, where Heath started his career. In 2010, the team relocated to Florida to become Orlando City, playing in the USL PRO division. In Orlando, Heath lead the team to two regular season titles and the league championship in their first two years. Heath was named USL coach of the year both years.