The vice president said in a recent interview that Dean was not the type of person to lead a political party and mentioned the chairman's mother.
"I've never been able to understand his appeal. Maybe his mother loved him, but I've never met anybody who does. He's never won anything, as best I can tell," Cheney said in an interview on FOX News Channel's "Hannity & Colmes."
Dean was elected governor of Vermont five times between 1992 and 2000. He ran for the 2004 Democratic presidential nomination but closed down his campaign after stumbling in the early primaries.
Dean has recently described the GOP as "pretty much a white, Christian party" and said many Republicans have "never made an honest living." Republican leaders have called on him to apologize, and even some Democrats have distanced themselves from his remarks.
Dean, in Boston on Monday for a fundraiser, told fellow Democrats that Republicans want a smaller government, but "their government is just big enough to fit inside Terri Schiavo's bed in the nursing home. We can do better than that."
Earlier this year, Dean, a physician, accused congressional Republicans of grandstanding in trying to keep the brain-damaged woman alive and said Democrats will use that against them in coming elections.
The party chairman also said on Monday that Democrats can win in traditionally Republican states.
"But we gotta be there and fight in order to do it. And believe me, we are going to fight back. I don't care if Dick Cheney likes my mother or not. We are going to fight back," Dean said to cheers and applause. "I think it's great that Dick Cheney went after me, to be honest. At least they notice there's a Democratic Party that's not going to put up with this stuff any more. So there's a lot we're gonna do."