A long-running battle over whether multinational corporations can avoid paying state income taxes shifts to the Supreme Court today in a case that could cost financially pressed states hundreds of millions of dollars.

The suit involves technical economic arguments, and the stakes have been high enough to attract more than two dozen friend-of-the-court briefs from the National Governors Association and Consumer Federation of America on one side to Sony Corp., Coca-Cola, Gulf Oil Corp. and Firestone Tire & Rubber Co. on the other.

Lawyers will debate today whether the Container Corp. of America must pay California about $75,000 in disputed income taxes from 1963 to 1965.

But the larger issue is California's controversial "unitary" method of collecting corporate income taxes, used in some form by 22 other states. Rather than examining how much money a company claims to make in the state, the unitary method in its broadest form looks at the firm's worldwide business activities and estimates what proportion of profits should be subject to state taxes.

Many corporations insist that this formula is unfair and that they should not have to pay state taxes on income earned by foreign subsidiaries.

But North Dakota tax commissioner Kent Conrad responded that "some corporations don't want to pay any taxes and look for every dodge, every loophole, every method to get an edge.

"It's just an accounting shell game," he said. "One of the things you learn in business school is to put your profits where there's the least tax burden. If you're a major multinational company, you can show your profits in a foreign country that has the least tax liability."

Attorney Franklin C. Latcham, who has been representing Container Corp. in the case for 14 years, said California's approach is unconstitutional. "California is taxing income earned in foreign countries," he said. "The method inevitably results in double taxation, by California and by the foreign countries involved. Every multinational in the world is vitally interested in this case."

Container, now a subsidiary of Mobil Oil Corp., has been paying the higher California taxes under protest since 1965. "There's a fair amount of money involved," Latcham said.

The Multistate Tax Commission, an alliance of 20 states, said a ruling against California could cost the states at least $625 million a year in lost revenue.

Container, which makes cardboard boxes and cartons, has subsidiaries in Colombia, Venezuela, Mexico, Italy and the Netherlands. Latcham said the company has little control over the foreign units and engages only in "arm's length" transactions with them. He said the subsidiaries are run by foreign nationals and that only 32 Americans have been transferred to them.

Opponents point out that Container provides 60 percent of the subsidiaries' financing and sells them $1 million a year in raw materials. Jim Hamilton, chief counsel of the California Franchise Tax Board, said Container should be viewed as one global company "as long as they're sending key people to their subsidiaries, training them, controlling their budgets, transferring know-how and selling equipment."

While some companies support unitary taxation because it lowers their tax bills, many have repeatedly challenged the system. In the early 1970s, for example, Mobil claimed a tax liability of $25 a year in Vermont during a period when it had sales of $9 million a year there. Vermont won a Supreme Court decision against Mobil, but only on narrow grounds.

"Multinational companies are becoming more aggressive in how they're characterizing their subsidiaries so they don't have to pay their fair share of taxes," said Paul Hanlon, Vermont's deputy tax commissioner. "All these companies we've never had any problem with are suddenly filing and claiming their subsidiaries are not directly involved with them."

In a similar court case last year, the Reagan administration angered many state officials by filing a brief that supported the corporate view against unitary taxation.

"I think the Justice Department was way off base in trying to trade away a state's right to obtain a fair and reasonable share of tax revenue," Vermont Gov. Richard A. Snelling (R) said.

The administration has been under pressure from foreign allies who claim that unitary taxes violate international trade treaties.

The multinationals also are lobbying hard in Congress, where Sen. Charles McC. Mathias (R-Md.) has been pushing legislation that would bar states from taxing foreign subsidiary earnings that remain outside the United States.

State officials say that without the unitary formula, based on payroll, sales and property, they would have to pore over corporate records to prove the connections between far-flung subsidiaries.

"It's simply an impossibility for us to go into a multinational corporation with hundreds of subsidiaries and try to recreate their transactions," North Dakota's Conrad said. "I've got a staff of 12, and I could spend a year auditing just one major multinational.

"It's a task that's beyond the ability of the Internal Revenue Service."