Spurs have now gone more than 400 minutes without finding the net in the league, having failed to score in all four of their games so far. Their attacking struggles were again evident as they created few clear-cut opportunities.
Tottenham came closest to breaking the deadlock in the fifth minute when Jan Paul van Hecke met Andy Robertson's free-kick at the far post, but his volley sailed over the crossbar.
Everton were marginally the stronger side and almost took the lead five minutes after half-time. Thierno Barry rose to meet a cross with a close-range header, only for Tottenham goalkeeper Antonin Kinsky to make a crucial save with his face.
Kinsky was involved again in the 65th minute after a poor pass out from the back handed possession to Kiernan Dewsbury-Hall. The midfielder attempted to capitalise, but the goalkeeper recovered quickly to keep out his effort with his leg.
The result leaves Tottenham 17th in the table, just one point clear of the relegation places. Everton, meanwhile, sit seventh and have preserved their unbeaten start after four league fixtures.
Alisson Becker has warned Liverpool they must find the "right mindset" to cope with the demands of a season across multiple competitions after a disappointing 0-0 draw with Fulham at Anfield.
Roberto De Zerbi has urged Tottenham to "stay calm" and maintain their confidence after their dismal start to the Premier League season continued with a 0-0 draw against Everton.
Arsenal maintained their perfect Premier League start to the season with a 2-0 win over Sunderland at the Stadium of Light on Saturday evening.
Chelsea were held to a 2-2 draw at home by newly-promoted Hull City, while Liverpool failed to score in a goalless stalemate against Fulham at Anfield in the Premier League.
