Victory would have seen Jose Mourinho's side leapfrog Manchester City, who had earlier recorded a 1-0 victory over Brighton & Hove Albion and they started off well as Harry Kane headed Spurs into the lead on 25 minutes.
Sergio Reguilon had delivered an inch-perfect cross and Kane powered home the opener.
However, as has become familiar of late, Tottenham failed to capitalise on their lead and Fulham grew into the game. The visitors eventually pulled level Ivan Cavaleiro headed home after great work from Ademola Lookman.
Reguilon thought he'd won it when he had the ball in the net on 89 minutes, but it was quickly flagged offside.
Jose Mourinho was frustrated at his for conceding late after scoring the opening goal and said after the match: "Once more we concede a goal that is completely avoidable.
"I have to admit in the second half they played more. They were losing and tried but Hugo Lloris did not make a save but we did not kill the game. Again we lose the man in the corner.
"We have lots of opportunities to kill off matches. In other cases especially away we are winning but then don't have that ambition to get the goal and kill the game.
"There are things that are individual. There are things that are individual qualities and individual mistakes. I cannot say much more than that."
A draw leaves Fulham in 18th place, while Tottenham stay sixth.
Information from Reuters was used in this report.