Sanchez was booed by the home fans when he appeared as a first half substitute for the injured Clement Lenglet, with Spurs leading 1-0.
The Colombia international was then sacrificed close to the hour mark to some more boos as Spurs chased the game after going 2-1 down.
Sanchez was at fault for Bournemouth's second goal while his replacement Arnaut Danjuma brought Spurs level at 2-2 before the home side conceded a late winner.
Spurs skipper Lloris was unhappy with the treatment meted out to Sanchez, who has been at the club since 2017.
The France goalkeeper beIN Sports: "It started earlier. It's when he came on the pitch. I've never seen this in my career.
"I feel really bad for Davinson. 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, for the fans, for the player. It's something you don't want to see in football.
Spurs' interim boss Cristian Stellini added: "I understand it's a tough moment for everyone and we have to analyse this moment."
"Everyone needs to analyse how important it is to support a player.
"I take the responsibility for the decisions I make. I thought it was too early in the game to use another striker because we were 1-0 up in the first half.
"When we were 2-1 down I thought it was the moment Davinson has to [come off]. It was only a tactical decision. We need to support him because it's a tough moment for him and also for all the team."
Tottenham have dropped to fifth on the Premier League table.
