The 6ft 6in forward was a half-time substitute at Hillsborough, but was sent off moments after the restart by referee Lee Probert for appearing to lead with his elbow in an aerial challenge with Wednesday defender Reda Johnson.
Smith was shown a straight red card and if the appeal is unsuccessful he will serve a three-game ban, starting with next week's home game against Leicester as Leeds bid to avoid a fifth straight defeat in all competitions.
"It wasn't a sending off," said McDermott, who revealed he had spoken with ref Probert about the dismissal after Saturday's embarrassing defeat.
"There's no way that's a sending off. Matt caught him but he was trying to make an honest challenge.
"I didn't think it was a booking quite frankly and the referee's got to look at it. They have to rescind it and if they don't, it's not right."
Leeds were trailing to first-half goals from Johnson and Atdhe Nuhiu when Smith was given his marching orders and McDermott's 10 men went on to concede four more as they crashed to their biggest league defeat in 55 years.