MORE AND MORE Washingtonians, it appears, are scrapping their oil-burning furnaces in favor of new gas appliances.

"My telephone has just rung off the book in the last two weeks with people wanting estimates," said David B. Fielder, who runs a heating/air conditioning business by the same name in Bethesda. "This is the first time I can ever remember in June receiving three or four estimates a day to change a furnace from oil to gas."

Behind the change in heating fuel preferences, at least im part; heating specialists say, is the steadily rising cost of oil prices. In a spot check of 11 Washington-area firms last week, most heating fuel retailers were selling oil in the 72 to 75 cents-per-gallon ragne with Amerada Hess, Inc. in Landover, Md., quoting the lowest, 67 cents, and Griffith Consumer Co. and affliliates the highest, 78.9 cents. Some heating experts talk of prices soon hitting 90 cents a gallon and perhaps even crossing the $1 mark during the coldest months.

The average gas customer uses about 1,500 therms a year, or $550 worth, said Susan J. Butz, Washington Gas Light Co. (WGL) spokeswoman. Heating with fuel oil, she said, is competitive with gas when oil is priced 45 cents a gallon. Last week, however, WGL asked for a 18 percent rate increase in the District of Columbia. Company officials expect that would raise the average monthly bill of a gas heating customer by $8.74, from $43.85 to $52.59.

"People are scared to death," said Tom Muse of Tom Muse Service Co., Inc., in Falls Church. Muse says he too is receiving about four calls a day for estimates on switching from oil to gas. Though he sees customers as concerned, not stampeding, "If you multiply it out, it's going to be panicksville again."

And while high prices may not bother some homeowners, many are clearly worried that oil will not always be available during the heating season. Federal officials fear a national shortfall and local distributors are making no guarantees.

"We didn't have any problems last year. But I'm going by what the news media are telling us," said Emma Bell of Washington Coal & Oil. "Who's going to stick their neck out and tell you you're even going to be alive tomorrow?"

"I don't think there's anyone in the world that can guarantee they're going to be able to deliver tomorrow," said David Fannon of Fannon Petroleum Services in Alexandria. "I wouldn't say we got plenty because we ain't got plenty. But we'll take care of our customers or die trying."

Switching from oil to gas heat requires removing the old oil furnace and tank and installing a new gas-burning furnace. Heating contractors say there are few complications, except, in some cases, finding an oil company to come out and drain the tank. Most contractors will install a forced air system, but some do not work on hydronic (warm water) heaters. You may need to call a plumber.

Installing a new furnace for hydronic heating will probably cost a little more than for forced air. Contractors estimate the total cost of a new forced air furnace as beginning at $1,000 and ranging to as much as $3,000 or more, depending on the size of the house. In most cases, they say, the price of a new furnace installation in the average-size home is running between $1,200 and $1,800.

Doubtful gas supplies forced the Washington Gas Light to cease accepting new customers from 1972 to 1978. Recently, however, the utility received permission to make a limited number of new hook-ups in the District, Maryland and Virginia, said spokeswoman Butz. WGL is allowed 8,000 new customers in Viriginia and 4,000 in Maryland through next spring. An initial test allocation allowed 3,000 hook-ups in the District. About 2,500 of the District's remain to be spoken for.

"We've been getting a lot of calls," Butz said. "It's moving at a steady pace, but there's still plenty of gas around. We're taking applications as we can process them."

Homeowners already with some gas service, such as a gas range or hot water heater, should call 750-1000 if they wish to install a gas furnace. New customers will have to be patient.

"It's not a matter of calling and we come out and put it in," she said. "It does take some time. They should get their requests in." Butz said it could take weeks, even months for gas service to be extended to new customers. "I can't give you any better indication than that. We're not giving out any time frame. We're just getting an awful lot of calls here and applications and they are being taken first-come, first-serve." Applications beyond WGL's allocation limit will be placed on a waiting list until next spring.

To request gas service, write the Washington Gas Ligh Co., Market Development, Springfield Operations Center, 6801 Industrial Rd., Springfield, Va. 22151. Include, if possible, the lot and block number of your home, your address and the type of equipment you plan to install. Except in rural areas, Butz said, there should be a gas main nearby. You can expect to pay around $100 for the initial meter and service charges.