Britt, 48, was trying to help students at the University Village Saturday morning when he suffered fatal injuries, Columbia Fire Chief Charles Witt said.
A second-floor interior walkway had begun falling when fire officials arrived, the Columbia Daily Tribune reported.
Britt was on the walkway when it collapsed and became entrapped. He was taken to the hospital, where he was pronounced dead.
Britt was a 23-veteran of the Columbia Fire Department.
About 75 people gathered for a candlelight vigil to honor Britt Saturday night.
"These are difficult times for all of us," Missouri University Chancellor R. Bowen Loftin said. "We are especially ... sending our condolences to Lieutenant Britt's family in this very difficult time."
No residents of the apartment were injured in the incident and residents have been evacuated.
Loftin said a thorough inspection of the school's buildings has begun.