The White House refusal to help Patricia Roberts Harris, secretary of housing and urban development, dump a bureaucratic antagonist has angered liberals and intensified the cold war inside the Carter administration.

Immediately at stake is whether Harris takes over the Federal National Mortgage Association - "Fannie Mae" - a federally chartered corporation that supports the housing market by buying commercial mortgages. But much more is involved. Backed by both civil servants and political appointees at HUD, Harris is a throwback to Great Society liberalism of the 1960s, which believed Uncle Sam and his checkbook could solve all problems.

Jimmy Carter, considerably less certain than Harris about most things, does not share that faith in government. There is, then, an internal conflict over both style and ideology between White House policymakers and departmental policy executors - particularly at HUD. But the president will not wield a hatchet. Carter is never inclined to fire anybody, and even it he were it certainly would not be the secretary of HUD. Black and female, Harris is fireproof.

Sytlistic differences overshadowed ideological differences when Harris's struggle with Fannie Mae began over a year ago. A prominent Washington lawyer before entering the Cabinet, she had served as a well-qualified director for several companies eager for public display of blacks and females. So she was shocked to find Fannie Mae's 10 stockholder directors were all white males.

It was not what Harris did next but how she did it. In her tough-gal style, she demanded that Oakley Hunter, chairman of Fannie Mae, integrate his board, sexually and racially. Hunter, a 61-year-old conservative Republican who served as a congressman from California 25 years ago, was non-plussed. Stunned by what he considered a personal insult, he gave not an inch.

The personality clash quickly became ideological. HUD claimed Fannie Mae under Republican Hunter was refusing to buy risky inner-city mortgages, as claim refuted by Fannie Mae and the housing industry. In no time, HUD had sufficient grounds to demand Hunter's ouster. The president himself agreed to look into it. Robert Strauss, the administration's peripatetic Mr. Fixit, tried to get Hunter quietly to resign. Failing that, one course remained: sack Hunter.

But to HUD's astonishment, the White House blocked the way by finding no grounds for firing Hunter. That recommendation came froma 32-year-old White House domestic policy aide named Orin Kramer who is now regarded as Benedict Arnold at HUD (in the words of one HUD official, "that little creep"). "I'm not going to let this turn into another Marston affair," Kramer told friends. "My job is to protect the president.

That desire is incomprehensible to HUD bureaucrats who couldn't care less about Jimmy Carter's political hide. Moreover, while HUD and Harris seek wider bureaucratic frontiers, in a style harking back to Lyndon Johnson by taking over Fannie Mae, Carter's men are more concerned about the housing industry in an uncertain economy.

The difference extends beyond Fannie Mae. The Harris team at HUD is frustrated by what it considers meat-ax budget slashes by Carter's Office of Management and Budget. "We are dealing with a bunch of conservatives, including a conservative president," a bitter Harris adviser told us. Carter aides believe HUD is living in a long-gone Great Society, submitting budgets that begged for the meat-ax.

Carter selected Harris for the Cabinet without knowing whether her style or ideology was faintly similar to his own. Having picked in haste, he can repent in leisure - for there is no chance of firing her. Her status among black leaders, organized labor and members of Congress wraps her in asbestos.

Accordingly, some worried Harris supporters see the White Hose refusal to dump Oakely Hunter as a prod to force out Harris. In fact, however, that presupposes determination of purpose and constanch of intent generally lacking at the White House.

Moreover, Harris has her own friends there. A few liberal presidential aides share HUD's position on both Fannie Mae and urban spending. Some softspoken Georgians assert an admiration for Harris abrasiveness. The president himself tries to convince key advisers how happy he is about HUD's secretary.

Nevertheless, the overriding mood among policymakers is impatience with Harris. In the latest bureaucratic infighting, HUD is issuing regulations to control Fannie Mae, and Fannie Mae is preparing a lawsuit to block the effort. An overburdened president, unwilling either to support or restrain his HUD secretary, looks on with unmistakable helplessness.