Bournemouth led at the break after a high energy first half, Tavernier finishing after some Milos Kerkez brilliance on the left to pick the Englishman out after a fine run up from the back from the Hungarian.
Evanilson gave the Cherries a two-goal advantage with a brilliant finish, before Pape Matar Sarr got a goal back for the hosts, his cross/shot finding the net off the far post.
Son Heung-Min then drew Spurs level to seal a point for the hosts, after he was brought down in the box by Kepa, the Korean converting from the penalty spot with six minutes to play.
Andoni Iraola made two changes from his side which beat Wolves in the FA Cup last weekend, as Ryan Christie was passed fit to start and Justin Kluivert returned to the starting XI.
Meanwhile, Ange Postecoglou made six changes to his side which lost in the Europa League to AZ Alkmaar in midweek, including the return of former Cherry Dominic Solanke to the team.
It was a rapid start from the Cherries, going close to opening the scoring twice in the opening five minutes.
Inside the first minute, Evanilson cut out a Romero pass out from the back and had his shot one-on-one saved by Guglielmo Vicario.
Then, the Cherries press again caused problems for the hosts, Kluivert this time played in after the ball was won high up the pitch, Vicario again getting down to save.
Kepa saved from a drilled Pedro Porro cross and the Cherries immediately counterattacked, Kluivert brought down on the edge of the D, but Marcus Tavernier’s free kick struck the Spurs wall.
The game continued at a high pace but with fewer chances for the two sides, as Solanke had a flicked header easily caught by Kepa.
Milos Kerkez challenged Brennan Johnson brilliantly after he was played in behind, timing his tackle to perfection to put the ball behind for a corner.
And Kerkez was at his brilliant best to set up the Cherries opener three minutes from the interval.
The young Hungarian intercepted a Porro pass in the Bournemouth half and surged into Spurs territory, having time to get his head up and put in a pinpoint cross for Tavernier to slide in and finish at the back post.
Spurs made a bright start to the second period but the Cherries thought they had doubled their advantage on 52 minutes, a rapid counterattack finished off by Kluivert, but Antoine Semenyo was offside in the build-up.
Hill headed a Christie cross wide after a neat short corner, before Kerkez drew a good save out of Vicario with a powerful effort that was aimed at the top corner from a tight angle.
Evanilson doubled the Cherries lead after 65 minutes, played in behind beautifully by Kluivert and lifting a wonderful finish over the Spurs keeper.
The hosts got a goal back two minutes later, though, Pape Matar Sarr’s cross/shot going over the jumping Kepa before hitting the far post and going in.
Vicario had to slide in to clear the ball almost off his own line under pressure from Evanilson, before the Brazilian and Christie were replaced by Dango Ouattara and Alex Scott.
Kluivert struck the post after a quick counterattack saw the Cherries have a three-on-three, Tavernier playing in the Dutchman, his effort not falling to Ouattara after hitting the upright.
Dean Huijsen headed over the bar and then just wide from Cherries corners as the game entered the final 15 minutes.
Tavernier made way for David Brooks after 80 minutes, before Spurs drew themselves level from the penalty spot.
Son Heung-Min was played in on the Spurs left and Kepa rushed out, going to ground as the Korean went away from goal, referee John Brooks pointing to the spot and Son converting.
Luis Sinisterra replaced Semenyo as the game entered stoppage time, with neither team able to find a winner and the spoils were shared.
The Cherries now sit eighth in the Premier League on 44 points.
Spurs: 1, G. Vicario; 4, K. Danso; 23, P. Porro (D. Udogie 87’); 24, D. Spence; 17, C. Romero (c)(M. van de Ven 61’); 8, Y. Bissouma (booked, 45’)(L. Bergvall 46’); 29, P. Sarr; 30, R. Bentancur (booked, 30’)(J. Maddison 61’); 28, W. Odobert; 22, B. Johnson (Son Heung-Min 46’); 19, D. Solanke.