Beeny's first screen role was in the film The Long Memory (1953). It stars John Mills and featured Thora Hird, Beeny's Future co-star in In Loving Memory and Last of the Summer Wine.

Beeny played Lenny Grove in the first British television soap, the BBC TV series The Grove Family, which was shown on Fridays from 1954 until 1957.

His big break came when he appeared in 47 episodes, as Edward Barnes, in the award-winning period drama Upstairs, Downstairs which ran for five series in the early to mid-1970s and was shown in more than 90 countries around the world. He went on to appear as Geoffrey in Miss Jones and Son; as Tony in the remake of The Rag Trade and as Billy Henshaw, Thora Hird’s co-star in In Loving Memory.

In 2001 he made a guest appearance in Last of the Summer Wine, something which he repeated numerous times until 2009 when he joined the cast as a permanent character. He appeared originally as the character Herman Teasdale, who later became Morton Beamish.

In 2006, he played a cameo role in Emmerdale. Coincidentally, he played the cousin of the character Noreen Bell, who had been played by his Upstairs, Downstairs co-star Jenny Tomasin. Beeny also played cameo roles in BBC TV's Sense & Sensibility and ITV's series Honest

Beeny's son, James, formed the band Virgin Soldiers in 2009 and has co-written a production that commemorates the 100th Anniversary of WWI. The production, The Dreamers, premiered at the Assembly Hall Theatre Tunbridge Wells in October 2014, and reached the West End in the summer of 2015. Beeny plays the role of The Storyteller in the production.

Beeny also toured in the play There's No Place Like a Home with Gorden Kaye, and in 2011 toured in Five Blue Haired Ladies Sitting On A Park Bench. He appeared in pantomime at the Assembly Hall, Tunbridge Wells in late 2011, early 2012.

Beeny's theatre appearances have included Stop the World – I Want to Get Off; The Long The Short And The Tall; Scapino; How the Other Half Loves; Night Must Fall; Oliver!; Who Saw Him Die; The Unexpected Guest; Boeing,Boeing; Move Over, Mrs. Markham; Bedroom Farce; Run For Your Wife; Rough Crossing; Present Laughter; Brief Encounter; Caught In The Net; Crazy For You; Ten Times Table, and Lark Rise to Candleford.