President Carter announced yesterday that he will not reappoint Arthur F. Burns as chairman of the Federal Reserve Board, and instead named G. William Miller, a New England businessman, who a virtually unknown to most of the official Washington.

Miller, 52, has been an executive since 1956 with Textron, Inc., the Providence, R.I. conglomerate. He has been president and chief executive officer, is a member of the board of directors, and since 1974 has been chairman of the board. He is regarded as a moderate-to-conservative on both economic and social policy, and has been active in several jon-creation programs sponsored by the business community.

It was not immediately clear whether Burns, 73, whose term as Fed chairman expires Jan. 31, would stay on as a member of the agency's seven-man board of governors. Carter said yesterday that he had invited Burns to remain on the board, but the outgoing chairman told reporters he would "have to think this one over carefully."

A board spokesman said Burns was expected to decide that question "shortly." If Burns chooses, he can stay on the board until January, 1984. He was appointed in 1970 for a 14-year term.

Burns, who was not notified about Carter's decision until a few hours before the announcemnet, praised the appointment as one "chosen wisely and well." He told Miller in a hand-shaking ceremony after the announcement: "you have my very best wishes and my respect."

Miller's nomination was obviously a gesture to the business community, which had lobbied almost unanimously to keep Burns on the job. Carter said he had checked Miller's appointemt with key business leaders and had received "expressions of confidence and approbation" from all of them.

Administration officials also suggested that Miller was chosen because of his experience in the international field. Textron does a substantial portion of its business overseas. Miller also has served for several years as a director of the Federal Reserve Bank of Boston.

The Miller nomination climaxed moths of waiting over what Carter would do about his fragile - and increasinly acrimonious - relationship with the Fed. Burns and the administration had been in a state of open feuding most of the fall.

The Miller appointment may be one of the most important Carter will make in four years in office. The independent Fed acts in effect as a fourth branch of government, setting basic money and credit policies that often can counter the administration's budget and credit policies.

White House insiders said it was Burn's continuing criticism of administration policy that finally led to the decision not to reappoint him. Top Carter administration economic officials had recommend unanimously against Burn's renomination. White House strategists had complained that Burns' anti-administration rhetoric often seemed to box them in.

The administration also had been upset over the Feud's recent tightening of money and credit policies. White House officials feared that any further restriction could abort the recovery - particularly if interest rates scored further than they have so far.

There was little indication yesterday how, if at all, Mille plans to change the policies Burns has set in force. Miller did little in response to questions except to say he hoped to "play a leadership role in continuing the policies Dr. Burns stood for so well and so long." Asked whether he planned to stress the fight against unemployment over the fight against inflation, he replied: "We need to work on both at once."

At one point in a question-and-answer session, Miller said he thought the Fed needed to "bring about conditions where the dollar is stronger."

However, in the next breath, he said be expected that "if we bring about the kind of normal progress in the economy we've been talking about over the next few years, the dollar will get stronger."

Burns appeared saddned and trying to make the best of it as he appeared with Carter and Miller at the announcement briefing. His voice cracked when he responded to the President invitation to say a few words about the decision. "I think," he told the group, "this is a good day."

There was no immediate indication how Miller stands philosophically compared with the conservative Republican Burns.

However, it was clear that in the arena of business community, Miller is regarded as something of an activist, and a moderate on social policy.

Miller was chairman of the National Alliance of Businessman, which operateds a summer-jons program for teenagers, and served during the 1963s on the President's Committee on Equal Employment Opportunity.

Carter stressed both these posts in announcing the nomination, saying he was "particularly impressed" with Miller's performance, "especially" his recognition of "the need to create new jobs."

Miller's record in public speeches and pronouncements also is relatively liberal. In a talk last January to the Traffic Club of Pittsburg, he backed a broad new economic stimulus to spur more jons in the economy, asserting that it would not impede the fight against inflation.

He also suggested the government should consider preparing "selective" government controls on the economy "as acceptable medicine under the circumstances" when "mushrooms of inflation" pop up. And he backed large new tax cuts for business.

op administration advisers seemed ecstatic about the appointment late yesterday. One policymaker described Miller as "tough, hard-driving, moderate-to-conservative. "He's just impressive as hell."

Burns, stepping down as chairman marked the end of an era of the nation's central bank. The Austrian-born economist made major changes in the Fed's approach to money and credit policy - adopting, and modifying, some of the "monetarist" ideas posed by University of Chicago Professor Milton Friedman.

Miller, dressed in a plain blue suit and accompanied by his wife, Ariadna, answered questions yesterday in a blunt and straight-forward manner, pausing to praise Burns at the point as "universally admired and respected."

If approved by the Senate, Miller would begin a 14-year term on the Fed as a representative of the Federal Reserve District of Kansas City - one of six "vacant" district from which Carter could have chosen, Miller was born in Sapulpa, Okla. His concurrent term as chairman would run for four years.

The chairman of the Federal Reserve Board is paid $57,500 a year. Other members of the board receive $52,500.

Miller's appointment came as a surprise to most observers here. Carter had been expected to choose from a list of well-known economist, including Paul A. Volcker, former under secretary of the Treasury for monetary affairs and now president of the New York Federal Reserve Bank.