Johnson has impressed in the opening half of the Premier League season and is the joint top-scorer at the Stadium of Light with four league goals.
However, Poyet feels his other players need to offer more in a creative sense.
"Sometimes we are a bit unfair with him (Johnson) and give him all the responsibility," the Uruguayan told the Shields Gazette.
"We need to share the responsibilities to go forward and create something. Someone needs to stand up, not always Adam Johnson.
"He was tired against Liverpool [in Saturday's 1-0 defeat]. Adam had been a father and hadn't trained for a couple of days. He wanted to play, but he had to track back too much.
"It was at the point where he couldn't go past people because he was so tired."
Johnson nearly added to his tally for the season against Liverpool with a swerving volley that thudded into the crossbar, having beaten Simon Mignolet.
