The symbolic poll shows the opposition Cheney faces from Republican leaders in Wyoming.
"This is what it looks like when the rule of law unravels," the Wyoming Republican wrote on Sunday.
"I think anyone with common sense would be reluctant to go talk to this committee," the GOP congressman said.
Representative John Katko, a New York Republican, announced on Friday that he would not seek another term this year.
The Republican congressman warned that "truth doesn't matter" to his political party.
The vice chair of the Committee investigating the Capitol attack said the House's top Republican is trying to "cover up" what happened by refusing to cooperate.
"What has changed is the use of conspiracy, the absolute lack of courage to call it out," the Illinois congressman said Friday.
The Illinois congressman said members of the party have to believe lies "to stay a Republican."
Cheney told Fox News viewers she was a conservative Republican, after being warmly welcomed by Democrats at a silence to mark the anniversary of the January 6 attack.
The Fox News host was unimpressed with the former vice president's comments about the Republican response to the Capitol riots.
Some Republicans condemned Donald Trump and his movement, while others attacked Biden and Democrats for focusing attention on last year's Capitol riot.
"It's not leadership that resembles any of the folks I knew when I was here," the former vice president said.
The Wyoming Republican said "history is watching" her GOP colleagues "and history will judge them."
The House panel investigating the January 6 insurrection has requested an interview with the Fox News personality.
"Simple decency would suggest that he pick another day," noted Douglas MacKinnon, who served under Reagan and George H.W. Bush.
"I think the country needs a strong Republican Party going forward, but our party has to choose," Cheney said Sunday.
"Not in any way true," the GOP Illinois congressman said.
On Wednesday, Trump's legal team filed a brief to the Supreme Court, asking the justices to rule in favor of Trump's executive privilege assertion.
House speaker Nancy Pelosi announced commemorative events marking the anniversary would take place. The ex-president will hold a news conference instead.
The Illinois Republican said the House select committee would "absolutely" be willing to subpoena members of Congress and former President Donald Trump.
"We sincerely hope other Republicans will join efforts to pursue the truth," the former lawmakers wrote in an open letter.
"As they echoed Trump, they primed the pump," former Fox News digital politics editor Chris Stirewalt said.
The Fox News host is calling for the Wyoming lawmaker's messages to be made public, claiming that she "begged" Donald Trump to pardon Scooter Libby, an aide to her father Dick Cheney.
Some members of Congress sent text messages to Mark Meadows discussing objections to the Electoral College votes.
Ingraham speculated that personal text messages from Representative Liz Cheney (R-Wyo.) and other 1/6 committee members would be "so aggressively boring."
Rep. Liz Cheney, vice chair of the January 6 committee, read out texts by the anchors as the violence played out at the Capitol.
The committee said Wednesday's deposition will go forward as planned, and that if Meadows refuses to appear, the panel will refer him for criminal contempt.
The number was revealed when committee chairman Rep. Bennie Thompson said the figure when pushing for contempt charges for Jeffrey Clark at a House hearing.
"This type of sadistic behavior can NOT be tolerated," Greene tweeted. "They must be held accountable for their reprehensible actions."
The Colorado congresswoman told Fox News which colleagues would be a focus if the GOP retakes the House.