Altidore, who joined Sunderland last summer, missed out on the Wembley defeat to Manchester City, having only scored two league and cup goals in 32 appearances.
The United States international featured for his country in the midweek friendly against Ukraine and could have the chance to lay down his marker on Sunday, when Sunderland face his former employers Hull in the FA Cup quarter-finals.
Although Poyet admits it was hard to omit Altidore, the Uruguayan coach is eager to see how the 24-year-old reacts.
"It was hard to leave him out but I needed to make the bench something I was comfortable with," he told the Daily Express. "I have a responsibility for the team. It's not nice, I don't like doing it but he has to cope with it.
"Let's see how he reacts. I look forward to seeing what type of character he is. I'd like him to prove me wrong. He should take his frustration out on the pitch.
"It's up to him. We saw it with Lee Cattermole. He is playing on Sunday but is that because of me? No, it is because of him. It's that simple. I would like Jozy to do the same - to show he deserves to play on Sunday and he will do well enough to play against Palace and Norwich. But because of him, not me. I'm not a charity."
Poyet concedes Altidore remained in his shell on the bench at Wembley but expects the striker to seize his opportunity when it is presented to him.
"His reaction on Sunday was quiet," he added. "He was professional but it was probably a shock for him because he didn't know. Only he knows how he felt inside.
"When these situations happen, you learn a lot from the players - how they take it. It's not a nice thing to tell someone who is not on the bench for a final but it's part of my job. You can't pick and choose which bits of the job you do but you hope they understand.
"It depends on the player how you react but if you get five minutes on the pitch, you can show them why they were wrong."
Despite dropping Altidore, Poyet insists the American still has a pivotal role to play at the Stadium of Light.
"We need him," Poyet admitted. "There is no doubt he still has a part to play.
"He's had too much pressure, he's come in with an idea he'll play for a manager and the manager changed and we changed the system. You can see he's got plenty of things to give but it's about the consistency and doing it all the time.
"To survive in the league you need a striker who will score double figures, if you don't you will struggle badly."