Redknapp may have worked wonders transforming the club following his arrival back in October, but his side are still languishing above the bottom three and have now failed to score in their last three league games at White Hart Lane.
It could have been even worse for Spurs had Andy Johnson not blown a golden chance right at the death when he was clean through on goal, but Heurelho Gomes produced a great save to earn his side a point.
Redknapp has already been linked with a move for West Ham striker Craig Bellamy and on the evidence of this display, they will need a number of new faces to ensure they pull clear of a relegation dogfight.
Fulham have now gone eight games unbeaten and Roy Hodgson must be thinking about challenging for a European spot come the end of the season.
Tottenham started the game brightly and could have opened the scoring on 26 minutes when Luka Modric showed all his class on the edge of the box.
The Croatia international superbly turned Danny Murphy and Aaron Hughes on the edge of the box and fired a left-foot which flew inches past the post.
Fulham keeper Mark Schwarzer produced a stunning save on 29 minutes to keep his side on level terms.
Benoit Assou-Ekotto burst clear into the Fulham penalty box and cut the ball back for Aaron Lennon, whose close-range shot was superbly kept out by Schwarzer diving low to his right.
From the resulting corner, Lennon crossed the ball into Ledley King whose close-range header flew just wide.
Gomes then matched Schwarzer for the save of the first half on 35 minutes when he denied Clint Dempsey.
Bobby Zamora's cross picked out the America international, who produced an spectacular overhead kick and the Tottenham keeper produced a superb one handed save to tip the ball over the bar.
The game was crying out for a goal and Fulham defender Paul Konchesky almost put his side into the lead on 57 minutes when he curled a free-kick just past the post.
Modric was clearly the dangerman for Tottenham and he had another golden chance on 64 minutes.
Assou-Ekotto's cross picked out the talented midfielder who found himself unmarked inside the box, but his close-range header flew inches wide of the post.
Fulham could have grabbed a dramatic late winner on 84 minutes when Simon Davies ball picked out Johnson, who found himself through on goal.
Johnson raced towards the Tottenham penalty area, but saw his shot superbly kept out by Gomes.