Schum attended the University at Buffalo, where he played for coach Jeff Quinn's Buffalo Bulls football team in 2010 and 2011 after transferring from Buffalo State College, where he played one season on their Division III football team.
After going undrafted in the 2012 NFL Draft, Schum signed with the Cleveland Browns on March 28, 2013. On May 13, 2013, he was waived by Browns.
On January 6, 2014, the Tampa Bay Buccaneers signed Schum to a reserve/future contract. He was waived by the Buccaneers on May 21, 2014.
Schum was signed by the New York Jets on June 19, 2014 to compete with incumbent Ryan Quigley in training camp. He was released on August 23, 2014.
The Buccaneers signed Schum to their practice squad on November 14, 2014, but was waived four days later.
The Jets signed Schum to a reserve/future contract on December 30, 2014. He was released on August 18, 2015.
On August 26, 2015, the Buccaneers signed Schum to compete with Karl Schmitz and Michael Koenen. By September 1, Schum was the last remaining punter on the Buccaneers roster after roster cuts. On August 28, 2016, Schum was waived by the Buccaneers.
Schum was claimed off waivers by the Green Bay Packers on August 30, 2016. He re-signed with the Packers on March 8, 2017. On June 1, 2017, he was waived by the Packers.