Spurs have failed to win any of their last three homes games in all competitions, including last weekend's 2-2 draw with Everton in the Premier League.
Dempsey believes that tension is coming into play at the business end of the campaign but has backed his side to show character and achieve big results under pressure.
He told The Sun: "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.
"Is it affecting performances? I don't know. It's been noticeable, in the last three home games. 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. 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."
With six games left to play in the Premier League, Spurs sit two points clear of rivals Arsenal, who have a game in hand.