He is known for his roles in television comedies, including Brass Eye, Big Train, Spaced, Jam, Green Wing, Friday Night Dinner, Upstart Crow, and Benidorm.

Heap was born in Kodaikanal, Tamil Nadu, India,[2] to an English father and American mother, the youngest of four boys.

[4] Heap starred in the BBC sketch show Big Train, where he performed a barefoot gymnastics routine and other sketches between 1998 and 2002, alongside other burgeoning comedy stars Simon Pegg, Julia Davis, Kevin Eldon, Catherine Tate, Amelia Bullmore, Rebecca Front, Nick Frost and Tracy-Ann Oberman.

[2] Between 2008 and 2010, he appeared as head postman Thomas Brown, in 32 episodes of the BBC period drama Lark Rise to Candleford.

[2] From 2011 to 2020, Heap played eccentric neighbour Jim Bell in the Channel 4 sitcom Friday Night Dinner, alongside Simon Bird, Paul Ritter, Tom Rosenthal and Tamsin Greig.

[6] He also played Andrew Thorogood in the BBC Four comedy Holy Flying Circus,[2] and Jonas in the eighth episode of E4's sci-fi comedy-drama Misfits in Series 3.

[2] He appeared as Robert Greene in three series of Upstart Crow (2016–18),[2] a BBC Two sitcom about Shakespeare, written by Ben Elton.

[1] From 2017 to 2018, Heap also appeared in the sitcom Benidorm,[1] playing the character Malcolm Barrett, the controlling and manipulative boyfriend of Pauline Maltby.

[2] In 2008, he co-starred in the surreal sci-fi B-movie spoof Captain Eager and the Mark of Voth, and was a publican in The World's End.

He starred in the music video for Four Tet's single "Smile Around the Face" in 2005, contributed a multitude of character voices in the audiobook "Do Ants Have Arseholes?".