Arlington County has lost its attempt to collect taxes on an apartment building that was bought by the East German government in 1976 to house embassy employes and their families.

The U.S. Supreme Court refused yesterday to consider the county's appeal from a lower court's ruling that the 30-unit building at 1515 S. Arlington Ridge Rd., once known as Windsor Park Tower, is immune from taxes.

Arlington County Treasurer Bennie L. Fletcher Jr. said he was disappointed at the outcome and would talk to the county attorney before deciding whether there is any possible future course of action.

Fletcher had sought to collect $55,000 in taxes for 1977 to 1979, the years covered by the legal dispute, and has filed liens for subsequent years.

In the District of Columbia, Congress has exempted embassies and embassy properties from taxation, as American outposts are exempt in foreign lands.

There is no similar law in Virginia. However, the United States and East Germany have agreed to recognize immunity for each other's diplomatic properties. Arlington County contended the federal government has no right to force a state to exempt property from taxation.

U.S. District Judge Albert V. Bryan Jr., in two rulings in 1978 and 1980, held that the East Germans owed taxes for the period prior to 1979, when the U.S.-East German pact was signed, but not for subsequent years.

The U.S. Justice Department took East Germany's side in appealing that ruling to the 4th U.S. Circuit Court of Appeals in Richmond, which decided last February against Arlington County. That was the decision the Supreme Court refused yesterday to consider.