In the show, participants race each other over hundreds of kilometres of harsh terrain with only what they can carry.
US white water river guide Spencer "Corry" Jones and team-mate Oliver Dev were disqualified after the incident.
In a clip from the show, Jones apologised on camera and said it did not sit lightly with him. He said he was desperate and hungry, but he knew he was breaking the rule when he did it.
In a statement, the Department of Conservation said it was clear with the production company that protected species could not be harvested or consumed. It said the company had alerted it about the incident shortly after it occurred.
DOC team lead of investigations Dylan Swain said the company was "on notice" about the need for programme participants to follow conservation rules.
He said there was a unique set of circumstances at play, that cast members were fatigued and suffering from significant hunger with an unusual group dynamic. He said this led the department to decide a warning letter was prudent.
The show airs on USA Network which said in an article on its website the pair were disqualified from the competition.
Weka are native birds and protected in New Zealand.
