Sanchez was at fault for Bournemouth's second goal to make the score 2-1 in the 51st minute, when he was turned inside and out by Marcus Tavernier. While he got a toe to the ball, the Colombian centre-back only succeeded in taking it into the path of Dominic Solanke, who beat Lloris to find the back of the net.
Tottenham fans booed Sanchez's next touch and interim manager Cristian Stellini decided to substitute him just seven minutes later, despite only bringing him on for the injured Clement Lenglet in the 35th minute.
"I've never seen this in my career," Lloris told beIN Sport when asked about the booing of Sanchez. "I feel really bad for Davinson because he's a team-mate, he's a friend and he's been fighting for the club for many, many years now.
"It's just sad. The story is sad for the club, the fans and the player. It's something you don't want to see in football."
Arnaut Danjuma, brought on to replace Sanchez, appeared to have rescued Tottenham a point with a fine low effort two minutes from time.
But Spurs were hit by a sucker punch when trying to find a winner. Bournemouth broke and Solanke teed up Dango Ouattara, who cut inside and curled home to spark wild celebrations from the away dugout.
Spurs are three points adrift of both Newcastle and fourth-placed Manchester United but have played a game more than the former and two games more than the latter. Tottenham face both teams in their next two fixtures.
"There is a lot of disappointment obviously," Lloris added. "It was more than a normal game because we missed a chance to go closer to Newcastle.
"And then obviously the story of the game, we went through different feelings and it's not easy to deal with but we have to recover from that."
As well as Sanchez, Spurs fans also directed their ire at chairman Daniel Levy with chants of "We want Levy out" ringing out from the South Stand in the second half.
Information from the Associated Press was used in this report.