It's been Winnie-the-Pooh, and profits, too, at Geico Corp. since John J. Byrne stepped in nine years ago to help save the insurance company from bankruptcy.

Yesterday, the ebullient chief executive announced that he will be leaving Geico and plans to move to California to head the Fireman's Fund Corp., the holding company of American Express' insurance operations.

"Winnie-the-Pooh is one of the great general managers of all time," said Byrne, adding that he admires the way the A. A. Milne character solves problems in a slow but steady way. "One should muddle about," said Byrne, who already has resigned as chairman of the board and chief executive officer of Geico but will continue for the time being as chairman of Government Employees Insurance Co.

"He's like a Knute Rockne in football. You don't forget a guy like Jack," said Warren E. Buffett, now Geico's largest shareholder. Buffett said he met Byrne at a party in 1976 and was so impressed by Byrne's knowledge of the insurance company that he was inspired to invest in the company. Buffett bought more than 500,000 shares of Geico stock the next day at $2.12 and $2.25 a share, an investment that paid off extremely well. Geico stock closed at $70.25 yesterday, down $1.25.

The 53-year-old Byrne has behind him a solid background of education in mathematics and experience at insurance companies. He received a bachelor of science degree from Rutgers University and a master's degree in mathematics from the University of Michigan. Before completing his degree from Michigan, he tried law for a semester at Harvard and also served in the Air Force.

Byrne began his career in insurance as a reinsurance actuary for Lincoln National Life, then helped found a company that later became the Life of America Insurance Corp. He joined the Travelers Insurance Co. in 1967, serving as a director of several subsidiaries and as executive vice president.

Soon after he arrived at Geico, Byrne began leaving his imprint. He cut the number of staff members in half and instituted annual information sessions that put managers on the spot in questioning sessions that could last as long as 15 hours.

Byrne complained that in the last two years his staff outgrew him as Geico found its feet and strode toward greater profits. "It's been almost embarrassing collecting my salary," he said with mock sheepishness. Although Byrne took a pay cut in 1976 to start at Geico with a salary of $150,000, the stock options and dividends written into his contract brought more than $2.4 million in 1980, making him one of the highest paid corporate executives in the nation that year. Last year, he received $419,000 in salary and bonuses.

Byrne is known for an unorthodox management style. Snyder noted that Byrne once told workers that if they couldn't meet a certain sales figure, six senior executives would be required to carry his 240-pound frame on a sedan chair into company meetings for a year. They beat the quota, however, so Byrne only got one ceremonial ride and had to pay his side of the wager in fancy dinners. "We're always doing dumb things like that," said the 190-pound Snyder. On another occasion when Byrne lost a similar wager, he cooked an Irish dinner for some Geico employes.

In another example of Byrne's practice of injecting his personality into company efforts, a current company goal is to cross on a chart the earnings-per-share figure with Bryne's golf handicap. The graph appears in the quarterly report to members of the board, according to Buffett. Byrne's handicap started out at about nine; the handicap is falling and the profits are rising, according to Buffett.

Byrne said he will present a more subdued image in his new post, adding that the operations of the Fireman's Fund are different from those of Geico. "Fireman's Fund is a much bigger company, a much more complex company. I will have to be more straight-up," he said. At Geico, "we lived through that awful awkwardness in '76. We made it a point to have a lot of fun. I don't think you can replicate that," he said.