The Jamaica international centre-back clashed with Michael Morrison and after consulting his assistant, Tanner issued a straight red card.
The official should have then awarded Leicester a penalty for the incident but failed to do so.
Nevertheless, Hart was left furious by his defender's actions and confirmed he would not be appealing the decision.
He said: "I don't think we can defend that at all. I've not asked him (Davis) what he was thinking.
"I will study what the code of conduct is, but my last word to the players is always stay on the pitch.
"It is the last thing we needed and very disappointing. It was tough enough with 11 players on the pitch.
"It's not for me to comment on what the players think but it was unnecessary and made the game difficult for us.
"Now I have to consider the recovery of these players because they have had to put in a real shift.
"But we played well in the second half. We were excellent and unlucky not to get a goal.
"We created chances in most of the games but all season the scoring has not been prolific.
"I will continue to back these players and help them as much as I can but we know we need some results."
Bruno Berner scored the only goal of the game in the 53rd minute to keep the Foxes on course for a top-six finish.
Nigel Pearson's side should have taken the lead in the eighth minute but Paul Gallagher scuffed wide.
Nolberto Solano then fired his free-kick from 30-yards narrowly over Julian Speroni's bar.
Palace striker Calvin Andrew came even closer in the 28th minute when his shot was tipped around the post by Chris Weale.
Leicester finally made the breakthrough when Palace failed to clear Lloyd Dyer's cross.
Speroni saved Martyn Waghorn's first shot but could do little to prevent Berner from slotting in the rebound.
The defeat leaves Palace hovering above the drop zone, while Davis will face another lengthy suspension.
Leicester assistant manager Chris Powell was pleased to seal the three points and keep the dream of a second successive promotion alive.
"It was a good win for us in our quest to finish in the top six," he said.
"You have to earn everything you get and to get three points in a feisty game is good.
"The sending off was significant and got the crowd on the backs of the officials. We were looking at it and wondering if one of our players would get sent off as well.
"But the manager (Pearson) kept us calm and said if we keep the pressure on we would get the goal and it came through Bruno."
Powell also felt Leicester should have been awarded a penalty following Davis' foul on Morrison.
He added: "The ball was in play for the penalty. It was from a free-kick and in the box so it should have been a red card and a penalty.
"Maybe the ref felt it was enough to send him off but by the letter of the law they should have given a penalty as well, but it is done now and we are pleased."