Spurs surrendered the initiative in the battle for fourth place on Sunday when they registered a 2-2 draw against Everton thanks to Gylfi Sigurdsson's 87th-minute strike.
Although they were seven points ahead of Arsenal last month, Tottenham will now slip a point behind the Gunners should they win their next game against Norwich on Saturday.
Injuries have played a big part in derailing Tottenham's Champions League hopes of late, but so has their form at White Hart Lane, where they have won eight times this year - compared to the nine they have registered on the road.
Spurs have collected the same amount of points on their travels as they have done at home this season and Dempsey admits White Hart Lane has become a tense place of late.
Asked about the tension, the Spurs forward said: "That's the way it's kind of been the last few games at home. It's been tense. I've felt like that's how the atmosphere has been.
"I don't know if it has been affecting performances. It's been noticeable in the last three home games (against Everton, Fulham and Basle). There are a few teams right there in the fight for the Champions League and it makes it more interesting for people to watch.
"It's a little bit more stressful but it's a good stress. At least we're pushing to try to get that spot. It's going to come to the end and, hopefully, we have enough character to get the job done.
"You've got to play. You've got to keep going and do everything you can to help the team. The race for a top-four finish will be a fight all the way until the end. We would have liked all three points against Everton but it was an important point."