Because of legal complications and a squabble with Congress, the United States today has no official tie to the Republic of China on Taiwan, and no office on Taiwan to serve the 5,000 Americans there.

This is apparently the most significant practical consequence of the formal opening today of full diplomatic relations with the People's Republic of China. Beginning today, the American and Chinese "liaison offices" in Peking and Washington are full-fledged embassies.

The old U.S. embassy on Taiwan, however, is closed, and the theoretically unofficial "institute" that is meant to conduct U.S.-Taiwanese relations in future cannot open for business because Sen. Ernest F. Hollings (D-S.C.) has refused to approve a transfer of State Department funds to pay for its operations. Hollings is chairman of the Apporpriations subcommittee which oversees the State Department budget.

Hollings, who disapproves of the terms of the Carter administration's decision to "normalize" relations with the People's Republic, said yesterday he thought it was improper to transfer government funds to the new institute before Congress has authorized the existence of such an organization.

The fact that the United States must close its offices on Taiwan today "is not my problem," Hollings said. "It's their problem," he added, referring to the administration.

As a result of the closing of the embassy on Taiwan, there will be no way an American there can replace a lost passport, swear a document for use in an American court or otherwise enjoy the protection of the U.S. government.

Also, there will be no one on Taiwan who can issue visas to Taiwanese or other foreigners to visit the United States. Even expeditious action by Congress to satisfy Hollings' complants will not rectify this situation for at least two to three weeks, administration sources said yesterday.

The administration believes all this could have been avoided if Hollings and his subcommittee colleagues had approved a "reprogramming" of State Department funds that would have transferred about $2 million to run the new institute.

"This is really irresponsible," one irate White House official said last night of Hollings' attitude. "This is one Carter is not going to be blamed for," the same source added.

Administration officials say they are concerned that irate Taiwanese may protest violently against the closing of the embassy, or that some serious legal problems will arise during this hiatus when the United States will have no official link to the Republic of China on Taiwan.

They say if anything like this occurs, Hollings should take the responsibility.

But Hollings declined to take any responsibility. He said yesterday that the hurry-up effort to establish a new legal basis for relations with Taiwan was entirely the administration'd idea.

[From Taiwan, special correspondent Michael Kazer reported that U.S. officials there let it be known several days ago that there would be no American office open for business "for 10 days" beginning today. Embassy officials -- many of whom will temporarily "resign" from the Foreign Service to work in the new institute, once it is established -- will remain in Taiwan and stay on the Foreign Service payroll.]

In another China development, the first U.S. ambassador to the People's Republic, Leonard Woodcock, was sworn in yesterday. Woodcock was confirmed by the Senate Monday, too late to reach Peking for today's official opening of relations.

At his swearing-in ceremony, Woodcock said he hoped to work for "a stable East Asia" and help build "a stable base for peace in the entire world." The 66-year-old former president of the United Auto Workers union has already spent two years in Peking as head of the U.S liaison office.

The House Foreign Affairs Committee yesterday gave final approval to its version of legislation establishing the new legal basis for Taiwanese-American relations. The bill must now be considered by both the House and Senate.

The Senate will take up the bill Monday and could act on it within a week. The administration hopes Hollings will allow the institute to open soon thereafter.

However, a State Department official said last night that it will take a week or more to prepare the new premises of the institute in Taiwan once money is authorized.