Raising his voice to a yell in a speech at New York University, Gore said: "How dare they subject us to such dishonor and disgrace! How dare they drag the good name of the United States of America through the mud of Saddam Hussein's torture prison!"
The Democratic former vice president said the situation in Iraq is spinning out of control.
"I am calling today for Republicans as well as Democrats to join me in asking for the immediate resignations of those immediately below George Bush and Dick Cheney, who are most responsible for creating the catastrophe we are facing in Iraq," Gore said, drawing strong applause from the partisan crowd.
"Donald Rumsfeld ought to resign immediately!" Gore bellowed. "Our nation is at risk every single day Rumsfeld remains as secretary of defense. We need someone with good judgment and common sense."
Rice "ought to resign immediately. She has badly mishandled the coordination of national security policy. This is a disaster for our country," he said.
The former presidential candidate was gentler on Tenet, a Clinton administration appointee, describing him as a friend and "honorable man" who should still leave his position for intelligence failures.
Gore said the abuse of Iraqi inmates at the Abu Ghraib prison was not the result of "a few bad apples."
"It was the natural consequence of the Bush administration policy," he said.
Gore said reservists photographed abusing prisoners "were clearly forced to wade into a moral cesspool designed by the Bush White House," which, he said, had abandoned the Geneva Conventions.
He said the crisis in Iraq has generated fierce anti-American sentiment and provided a strong recruiting tool for terror groups.
President Bush "has exposed Americans abroad and Americans in every town and city to a greater danger of attacks by terrorists because of his arrogance, willfulness and bungling at stirring up hornets' nests that pose no threat whatsoever to us," Gore said.
Gore said that because of the war, Iraq has "become the central recruiting office for terrorists."
The administration, he said, has also set up U.S. soldiers for "payback the next time they are held as prisoners."
The speech was one of several Gore appearances sponsored since August by MoveOn.org. The liberal interest group also has a television and radio ad calling for Mr. Bush to fire Rumsfeld.
Gore, who served in Vietnam, predicted greater problems for America's involvement in Iraq. "The worst still lies ahead," he said.
Gore said electing Democrat John Kerry was the first step toward dealing with Iraq.
He said that Kerry should not "tie his own hands" while campaigning by offering any specific proposals for how he would handle a situation that is "rapidly changing and, unfortunately, rapidly deteriorating."
The White House issued no reaction, reports CBS' Allison Davis. Instead, the Republican National Committee issued a paper statement. Communications Director Jim Dyke said the following:
"Al Gore served as Vice President of this country for eight years. During that time, Osama Bin Laden declared war on the United States five times and terrorists killed U.S. citizens on at least four different occasions including the first bombing of the World Trade Center, the attacks on Khobar Towers, our embassies in East Africa, and the USS Cole."
"Al Gore's attacks on the President today demonstrate that he either does not understand the threat of global terror, or he has amnesia."