Burr said Tuesday’s announcement had been a shock but not necessarily a surprise, and he believes Millington will be missed.
"The club have lost a very good man in Grenville because he put the club first," said Burr.
"The club was the most important thing to him and he understood the football side of things and what we’re up against in this division.
"He would come down to training as much as he could to watch the lads and he had a lot of passion for the club. For me the club have lost a really good man and I’m disappointed.
"The one thing Grenville Millington did was put the club before everything else every single time.
"It came as a shock to me but then again at the same time perhaps not so much of a surprise.
"I spoke to him after the Gateshead game and I could see he wasn’t his usual self.
"Sometimes when you’re not winning matches or have lost a game it can be difficult to pick yourself up but Grenville was always very positive and he understood how difficult it can be in this division and the level we are at.
"He was always very supportive and I’m really disappointed that he’s taken this decision but I respect his decision."
Millington’s departure means the fan-owned club is now looking for its fourth chairman since reforming in 2010 and his successor is expected to be announced after the board meets on December 16 with vice chairman Jim Keegan taking charge in the interim.
Chester, who dropped to 15th in the National League with a 1-0 defeat at Gateshead last time out, visit leaders Cheltenham Town on Saturday.
Ben Heneghan misses out as he must serve a one-match suspension having reached five cautions for the season so captain Ian Sharps should return to partner Alex Whitmore in central defence.
The sides drew 1-1 at the Lookers Vauxhall Stadium back in August but the Robins will be full of confidence having gone 10 games without defeat.