Federal Reserve Board Chairman Arthur F. Burns made what he called his "final public appearance" as head of the nation's central bank yesterday, with a warning that government has failed to cope with serious "underlying" economic problems.

Burns addressed the National Press Club - his only appearance there - and told the audience of newsmen that "it is not beyond the realm of possibility that I may be joining your ranks."

Sources said that Burns' reference to entering the ranks of journalism was a serious one. He has had many offers to write and comment on public affairs, some of which he is considering. He also has received numerous offers from business firms, and from academic and other institutions.

The 73-year-old economist, who has served as Fed chairman for the past eight years, is scheduled to retire today. Textron chairman G. William Miller was named as his successor by President Carter.

But Miller's confirmation apparently will be delayed for a brief time by the Senate, while its banking committee checks into commissions paid by a Textron subsidiary in connection with a large sale of helicopters to the Government of Iran.

Under federal law, the incumbent Federal reserve chairman stays in office until his successor is formally sworn in. In any event, Burns has agreed to continue as one of the seven board members until March 31 to help smooth the transition.

Burns did not attempt yesterday to break any new ground, or bring up points of contention between himself and the Carter administration. He was cordial, as well, to the press - despite some criticism he has received over the years - expressing "great admiration for the professional job" done in covering economic affairs.

He did offer a mild jibe, however, at "broadcast journalists, who are not as effective as they might be" on economic coverage.

With Treasury Secretary W. Michael Blumental seated next to him at the head table, Burns reiterated his strong feelings that the recent "sharp decline" in the value of the dollar could have serious consequences for the domestic economy and the United States' relationships abroad.

"Much of the conventional wisdom had argued that a depreciating currency will help rather than hurt a nation's trading position," Burns said. But he suggested that continued uncertainty about the future of the dollar "could produce a disorderly flight out of dollar assets, and that could be inimical to domestic prosperity."

Burna said that the administration recognizes the urgency of the dollar problem, and had taken steps recently to protect its value. Later, in answer to a question, he remarked that an increase in the discount rated voted by the Fed at the end of last year to protect the dollar had "helped confidence in the dollar around the world" and had not hurt the domestic economy.

Administration officials have remained silent on this issue but are known to feel that increasing the discount rate to defend the dollar might better have been held in reserve. In his confirmation hearing last week, Miller refused to criticize that specific action of the Fed, but indicated he might have taken a different position.