The Arlington County Board gave preliminary approval yesterday to plans for a bond referendum needed to finance the county's $40 million to $50 million share of completing the 101-mile Metro subway system.

A decision on the exact size of the bond issue probably won't be made until early next year after regional officials calculate how much money the county's share will be.

Arlington thus became the third Virginia locality to approve the plan for completing the system. Earlier this week the cities of Falls Church and Alexandria endorsed the completion of the subway system.

The Arlington action yesterday could be reversed if the political balance on the board shifts to a Republican-endorsed majority after the Nov. 7 elections. Should that occur Walter L. Frankland Jr., an outspoken critic of Metro, could beome the board chairman. Frankland, some Metro supporters say, would lead an effort to overturn yesterday's vote.

Franklin yesterday was the only member of the board to vote against the major provision of the local Metro plan.

Federal officials have said that the Carter administration supports the goal of the 101-mile system but local governments like Arlington must first guarantee payment of their share of construction and long-term operating costs.

In a series of briefings to local governments, Metro board chairman Joseph S. Wholey, also a member of the Arlington board, has warned that should any jurisdiction refuse to pay its share, "the system would grind to a halt."

The plan Arlington endorsed calls for completing the subway system by 1985, with about 80 percent of the $3.2 billion completion cost paid by the federal government. The remaining 20 percent would be paid by state and local governments.

Arlington officials noted that their referendum depends on guarantees from other localities and on the effect of tax-cutting proposals similar to Proposition 13 that will be decided by voters in Montgomery and Prince George's counties.

Several residents testified against completing the system. "Belief in Metro has been raised to something of a religion," said Liberatarian Party candidate Richard Gardiner who is running for Wholey's seat on the county board. "But I think politicians are creating a tax monster. Car transport is superior in every respect to mass transit."