Tigers coach Brian Kelly told reporters after the game that Burns will miss the next 3-6 weeks recovering from the injury.
"I don't want to get into specifics when you're talking about a neck injury, but he's been cleared by all the doctors," Kelly said. "He needs some rest."
Burns entered the weekend second on the team with 17 tackles. His 2.5 tackles for loss were tied for the team lead.
Even without two starters, LSU's defense pitched a shutout against the Lobos, giving up only 88 yards of offense
Quarterback Jayden Daniels, who threw for 279 yards, left the game in the third quarter with an injury for the Tigers (3-1).
Kelly said Daniels strained his lower back and could have returned to play, but the game was already in hand.