Vincent C. Burke Jr. announced yesterday he will resign as chairman of Riggs National Bank to join a local law firm as the District's biggest bank undergoes a high-level management shakeup.

Three other senior officials have left the bank or have been reassigned since September when Riggs' biggest stockholder, Joe L. Allbritton, named himself chief executive and stripped Burke of his main responsibilities.

No replacement for Burke, 60, was named yesterday. Further changes in top management at Riggs bank are expected when the bank's board of directors holds an organizational meeting Wednesday.

At least one Riggs director voiced concern yesterday that Allbritton, who bought control of the bank in April 1981, may be decimating Riggs' top ranks. "It's one thing to bring strong management to a company. It's another thing to tear a place apart," said the board member, who asked not to be identified.

Changes in top management leave in doubt the future of bank president Daniel J. Callahan III, 50, the chief operating officer of Riggs, who has worked in tandem with Burke for nearly seven years. Callahan did not return a reporter's calls yesterday.

Speculation about who will be promoted in the shakeup centered on Executive Vice Presidents William H. Ryland and William G. Tull. Ryland, 48, who headed the bank's London operation, was brought back to Washington last year to oversee the bank's problem-plagued corporate lending operation. Tull, 50, is the bank's chief financial officer. Both declined comment yesterday.

Sources in the Washington banking community said several other senior executives at Riggs may leave.

Since Allbritton bought control of Riggs, top officials at the bank have expected that the Texas financier would make major changes in the bank's management. When Riggs' profits began to slip in late 1981 and 1982 as a result of growing problem loans to foreign borrowers and real estate developers, there was increased speculation about Allbritton's ordering management changes.

Riggs is expected to announce its earnings for 1982 in the next few days. Profits for the first nine months of the year were down 16 percent to $15.6 million from $18.5 million for the first three quarters of 1981.

The changes in the $3.9 billion bank's management began in September, when Allbritton took the title of chief executive away from Burke. At that time, James D. M. McComas, head of the bank's international department, resigned to set up his own consulting firm.

Last week, Webb C. Hayes IV, head of the bank's corporate lending division, resigned to become president of Palmer National Bank, a newly chartered bank that expects to open its doors here in May.

William Moreland, head of Riggs marketing division, was reassigned last week. A bank official said he did not know Moreland's new assignment.

Burke, in announcing he would leave the largely ceremonial post of chairman of the bank, said he would join the Washington law firm of Steptoe & Johnson to specialize in the changing laws governing financial institutions.

Allbritton, who owns WJLA-TV and other broadcast properties and formerly owned The Washington Star, won a bruising takeover battle for Riggs in 1981. After paying $70 million for about 40 percent of Riggs' stock, Allbritton formed Riggs National Corp., a new holding company that owns only the bank