Corr was born in Melbourne. He started his acting career at the age of thirteen with the film Opraholic. His first television performance was on The Sleepover Club with a supporting role as Matthew McDougal. After completing filming on The Sleepover Club, he scored a lead role as Sheng Zamett on Silversun. Soon after, Corr had many guest appearances on shows such as Scooter: Secret Agent, Blue Heelers and Neighbours. Corr is also a voice actor and was the voice of a sheep in Charlotte's Web. In 2006, the second series of Blue Water High aired on Australian TV in which he had a lead. Corr starred as a young Surfer named Eric, who was one of the six chosen teens to attend Solar Blue. In 2009, he graduated from the National Institute of Dramatic Art (NIDA). In 2010, Corr appeared in the final few episodes of Underbelly: The Golden Mile as Michael Kanaan and was a regular character on the popular Australian television show Packed to the Rafters, playing Dave Rafter's nephew Coby Jennings.

Corr was the 2011 recipient of the Heath Ledger Scholarship. He was a nominee for Cleo's Bachelor of the Year in 2012. In 2013, he appeared in the teen thriller film 6 Plots, and starred alongside John Jarratt in Wolf Creek 2, the 2013 sequel to the horror film Wolf Creek.

In May 2014, Corr was charged with heroin possession after being found with a freezer bag holding 0.26g (less than 1/100 of an ounce) of the drug in the Sydney suburb of Bondi. After admitting to the possession, the actor was given a 12-month behaviour bond with no conviction recorded. The arrest took place only a few days after Corr’s grandfather (who flew with him when traveling to acting jobs as a child) had gone into a coma following a heart attack. In a “vulnerable state,” Corr went into a hotel, where a man gave him a bag containing heroin and said, “here, have this, you’ll feel better.” According to Corr’s Lawyer, Corr was grateful for having been arrested, because it stopped him from using the drug, “which he said he had never tried before.” As Corr later stated, “I found myself in a position where I was incredibly upset, someone offered me something and 200 metres after, I was picked up with it. It was an error of judgment.”

In 2015, Corr starred in Neil Armfield's romantic-drama film Holding the Man, adapted from Timothy Conigrave's 1995 memoir of the same name. Corr portrayed Timothy Conigrave opposite Craig Stott who portrayed John Caleo, with supporting performances from Guy Pearce, Anthony LaPaglia, Sarah Snook, Kerry Fox and Geoffrey Rush.