Former first-team coach Steve Davis and goalkeeping coach Phil Hughes have also now both left the club.
Burnley have confirmed that interviews are "ongoing" for the manager's job, with Doncaster's Sean O'Driscoll and former Sheffield Wednesday boss Brian Laws both linked.
Huddersfield boss Lee Clark has dismissed reports he was interviewed by Burnley chairman Barry Kilby and operational director Brendan Flood yesterday.
Dobson told club's website: "We are taking it a day at a time because we don't know how long it will last.
"It could be that a new manager can be in before the game on Saturday so all we can do is say, 'Right, let's make sure the training sessions are correct, the players apply themselves the right way and then everything else will take care of itself'.
"I was delighted with the session this morning. 'Pash' put on a terrific session, along with the senior players.
"The players all applied themselves exceptionally well, their attitude was terrific and if we get that response throughout the week I'll be well pleased."
Dobson, a former Clarets midfielder and England international, promised to keep the players updated with any developments.
He said: "I spoke to the players in the dressing room and we've tried to clarify everything. I know from being a player if something like this happens it's unsettling for them.
"They want to be finding out exactly what's happening for the football club and any developments at all. I reassured them anything that I know gets passed onto the players."
The club hope to have a new manager in time for Saturday's trip to Manchester United in the Barclays Premier League and Dobson insisted the players will focus on nothing but the game itself.
"It's a case of knuckling down now and taking this on the chin. What's gone is gone," Dobson said.
"We'll do whatever we can. We can only manage the things within the football club.
"Whatever's going on outside is someone else's responsibility.
"I just clarified that with the players and reminded them that they have shown a tremendous attitude in the last two years.
"They deserve to be in the Premier League and I'm sure one of the reasons they were delighted by the promotion was travelling to places like Old Trafford and playing against the very best.
"That's what they're doing this weekend, so we'll focus on that."