Smith was a half-time substitute for Tom Lees at Hillsborough but was shown a straight red by referee Lee Probert for an aerial challenge on Wednesday skipper Reda Johnson within a minute of the restart.
Brian McDermott told the club's official website: "I can't understand why it wasn't overturned. I'm sure they will have their reasons, but there was no intent.He's gone in, in a fair way, he bumped into the player, and he went down easy.
"I don't believe the fracas afterwards was helpful with players from Sheffield Wednesday running a long way to get involved. I'm very disappointed with the outcome."
Smith will serve a three-game ban and will miss the next three games against Leicester, Ipswich and Huddersfield, all at Elland Road.