The American Petroleum Institute, the chief lobbying arm of the oil industry, has laid off 35 employes, or 10 percent of its Washington staff, as a result of belt-tightening efforts by the oil industry precipitated by plummeting oil prices.

A spokesman for the trade association said that this was the first time in the group's 67-year history that it has been forced to take such drastic measures in response to industry problems. "The whole industry is cutting back. We felt it would only be right to make cuts in our staff as well," said API spokesman Joseph Lastelic.

Over the past three months, crude oil prices have dropped from $27 a barrel to less than $10, hitting their lowest level in three years. The result has been massive layoffs and cutbacks by oil firms nationwide.

The decision to lay off staffers and cancel programs at the American Petroleum Institute was made after a mid-March meeting of the group's management board, which includes both association executives and corporate members. At the meeting, the board slashed the institute's $50 million budget by $3.6 million.

Not only were major research projects on health and environmental issues terminated, but the board also canceled its newsletter on oil exploration and a monthly statistical release on liquified petroleum. It also deferred professional salary increases indefinitely and eliminated out-of-town meetings.

Another Washington oil industry association, the Independent Petroleum Association of America, has been paring its staff and cutting expenses since the first downturn in the industry in 1981, said its president, Lloyd Unsell.

"Recently, we even counted the number of phones that our association uses and decided that we could do without 13 of them," Unsell said. The group, which represents 6,500 independent oil producers in the United States and Canada, also has been forced to fire three support staff personnel in the past five years.

"We have an industry out there that is reducing its work force by 25 to 30 percent. In times like this, dues to industrial organizations such as ours become very vulnerable," Unsell said.

Other Washington associations representing the oil industry, such as the American Gas Association and the National Petroleum Refiners Association, have not had layoffs, mostly because their staff sizes were very small to begin with, association sources said.