Seeking an eighth straight league win to keep up with the class of 1897, the Clarets went two goals down in the first half as Scott McDonald and Danny Shittu both struck for the home side.
Sam Vokes notched his ninth goal of the season just a minute after Shittu's deflected strike however and Shane Lowry's own goal on 55 minutes handed Burnley a share of the spoils.
"You can't go in every week thinking it's going to be easy just because we are top of the league and I don't think we do. We certainly didn't here. I played here and I know it's not easy," said Dyche.
"But there is 11 people out there every week that are trying to stop you - good players and good teams and good clubs.
"Anyone at any time can lose games or have a tough run and we know how it works so I certainly won't be worrying if we have a few results against us. But there is a challenge there for us to keep going.
"Their second goal sprang us into life, we couldn't get out of a fog in the first 30 minutes and then the goal brought us into life and we literally immediately, right in front of your eyes, looked full of life again like we have done all season.
"I told the lads 'be ready because these boys don't lie down'. No teams lie down and certainly not Millwall because the crowd won't let you lie down here."
For Millwall it was third straight draw but the Lions have now not won in the league since late September, and boss Steve Lomas admitted his side needed to maintain their two-goal advantage until at least half-time.
"I suppose when you go 2-0 up it's a delicate scoreline and we wanted to try and get to half-time 2-0 up but the goal has gone in," he said.
"It's a little bit of misfortune but the second goal was poor. We need to defend better but we didn't and we paid the price.
"We are all disappointed but the level of performance in the last three games has been very good. But for small margins and a little bit of luck and a few decisions to go our way which we haven't really been getting we are not far away.
"We have got to take the positives from it because performances are a lot better and we are fighting and scrapping for each other.
"Scott (McDonald) has scored goals wherever he has been and he is a quality striker and that's what he was all about so I am delighted for him."