They do at Indians games. They play hang on sloopy and do the Ohio chant. They also have the Ohio letters they give to fans and show them on the scoreboard.
The Monsters used to acknowledge Ohio State-Michigan rivalry, but haven't lately. They would have done something last night during the game but I don't remember anything being mentioned.
Don't know about Browns or Cavs games, but it wouldn't surprise me. I did just remember when I went to games in the 80-s and 90's, the Browns would have the OSU marching band once a year. Not sure if they still do or not.
Brutus literally threw out the first pitch at the last Guardians game I attended and have definitely heard Hang on Sloopy at Browns games
Just offering my anecdotes