Arsenal's home win over Norwich City on Saturday demoted Spurs to fifth spot and the North London rivals both have six games left to play.
Dawson remains optimistic of qualification for the Champions League and vowed there will be a lot of 'hard work' over the final weeks of the season.
"We're not thinking of that," Dawson told Sky Sports News when asked how disappointing it would be to miss out on the Champions League.
"As players you can't. Who knows what will happen on 19th May? Hopefully it will be a good day and we're celebrating. There's a lot of hard work to go in from ourselves and also the other teams to get there.
"Arsenal have gone above us and we're in fifth. The games come thick and fast and, with us not playing at certain times with the way they've fallen, we've just got to concentrate on what we do.
"That's all we can affect and, with six games to go, we hope to pick as many points up as possible to get us back in the top four."
The centre-half also rejected suggestions that the Europa League had impacted upon Spurs' Champions League push after Andre Villas-Boas' side lost on penalties to Basel in the quarter-finals last week.
He added: "I don't think it has at all. If you look at our results after the Europa League, I think they've been really good.
"It's just that, after getting as far as we did, we wish we could have gone all the way. It would have been great to be lifting that trophy, but that's not going to be, so we'll be concentrating on the Premier League now."