The game sparked into life in the second half when Nacer Chadli fired home for Tottenham on 52 minutes following a Ben Davies cross.
However, Burnley levelled through Sam Vokes, who was making just his second appearance of the season since recovering from a knee injury that had kept him out for nine months.
The winners of the replay will face a home tie against Leicester City in the fourth round.
On-loan Manchester United defender Michael Keane was given permission to play for Burnley while
full-back Daniel Lafferty made his first start of the season in place of injured captain Jason Shackell.
Tottenham boss Mauricio Pochettino made seven changes from the team that stunned Chelsea on New Year’s Day as goalkeeper Michel Vorm and striker Roberto Soldado were among the starters.
There was a low-key atmosphere in the stands with only three sides of the ground open and it transferred to the players on the pitch as neither side produced anything of real note during a distinctly lacklustre first half.
The only shot of the half came when Christian Eriksen picked out Paulinho on the edge of the box but his well-struck volley was blocked by Kieran Trippier.
Pochettino had clearly been underwhelmed with his team’s performance in the opening 45 minutes and unleashed Harry Kane into the action at half-time.
The striker replaced Benjamin Stambouli and Spurs upped their intensity.
Tom Heaton was brought into the action by a header from Federico Fazio on 51 minutes, but it was an easy save for the goalkeeper.
The goal did come five minutes later through Chadli.
Kane picked out Ben Davies on the left and his cross was controlled neatly by Chadli, who shot past Heaton with the aid of a deflection off Mee.
Spurs almost had a second moments later as Kane broke into the box but Heaton stood up to block his effort.
Dyche sent on Vokes, who has had to be patient following a brief comeback appearance against Liverpool on Boxing Day following his injury nightmare.
He injected life into the crowd and his teammates as the pressure started to grow on the Spurs backline. Vokes’ moment came with 17 minutes remaining.
The striker was perfectly positioned in the centre of the box to guide Michael Kightly’s cross into the far corner before being mobbed by his team-mates.
Tottenham came closest to a winner, with Mee blocking from Eriksen and Kane screwing wide, but they had to settle for a replay at White Hart Lane.