Eyre moved to Spotland after a 21-year spell at Manchester City working with the side's youngsters and won five youth titles and the FA Youth Cup during his time at the club.
But despite his League One side gaining a reputation for an attractive, passing style of football, Rochdale have struggled this season, scoring just 22 goals and winning only four of their 21 league games so far.
"I do have nice things to say about Steve Eyre and Frankie Bunn, they're nice people and they worked very hard for the club," added Dunphy.
"[But] if we carry on this form throughout the season we'll be facing certain relegation.
"We currently have a squad of 28 players which is the biggest Rochdale Football Club's ever had by a long way - Keith's squad was about 18 or 19.
"We're also on the biggest budget we've ever done at Rochdale, which is not a fortune but it is a big clump.
"You add them all together and I feel that I've done everything I can as chairman to support the management team, and if we're not getting results the only thing I can do is change the management team."
Youth team coach Chris Beech will take temporary charge of the club, with Rochdale's next fixture coming against Tranmere on Boxing Day.
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