He probably thought they would get thrown out and nobody would see them. He didn't know about the janitor's vengence.

It was pure hubris on Mr. M's part. He had come to believe he ruled the school, he looked down on the custodian.

The option I always thought was correct was to simply change one vote to Paul.

Remember, he was "teacher of the year" a number of times, and a guy who preached about morals & ethics.
He wasn't experienced, in other words, with "doing the wrong thing."
So of course he made a dumb mistake trying to cover up an impulsive action that, for him, was already very much "out of character."
He was not a smart thief, he was a good teacher who was already having a really bad day.

It would have been smarter for him to do something more elaborate, but it's probably one of the movie's way to nod to the 90's, a recent but utterly different era then today, where things were more innocent and you did not have to think things through so much. If I were in his position and doing that today, I would have to search the room first to make sure nobody hid a pen or cell phone camera to film the vote counting, then I would have to smuggle them out of the school and burn them.

I thought the same thing when I first watched it, but I figured he threw them in the trash to quickly get rid of them (the 2 students were sitting across from him and watching like a hawk). Then he could take them out of the trash on his way out and dispose of them, but he probably forgot since he had so much on his mind at that point. that might be a possibility?