Fuel prices continued their relentless upward climb yesterday as Exxon, the nation's biggest oil company and the source of one-fourth of the gasoline in the Washington area, raised the price of gasoline another nickel a gallon.

At the same time, Exxon announced a 3-cents-per-gallon price increase for home heating oil -- instantly triggering a 4-cents-per-gallon increase at Metropolitan Fuels Co., a large Bethesda distributor that depends entirely on Exxon for its supplies.

Metropolitan president Tom Curtis said the extra penny per gallon was to pay "operating costs" such as carrying expensive oil inventories on money borrowed at high interest rates.

Yesterday's Exxon price increase is the latest in a stunning series of increases by all the nation's oil companies that began in mid-December as the Organization of Petroleum Exporting Countries met and raised crude oil prices dramatically.

The average price of a gallon at heating oil in the Washington area yesterday was 95.3 cents, up 5.1 cents in just five weeks, according to a Washington Post survey of 15 heating oil distributors in the area.

Gasoline prices have risen even more drastically.

Yesterday's Exxon increase, for example, comes on top of 3- and 6-cent increases -- a total gasoline price increase of 14 cents a gallon, or 21 percent for Exxon since mid-December.

Since the start of 1979, Exxon's gasoline prices have increased 36 cents a gallon, or about 79 percent.

In addition:

Shell, with 6 percent of the gasoline market here, went up 10 cents a gallon on gasoline in just nine days early this month, making a 95 percent price increase since the start of last year.

Amoco, with 26 percent of the market here, went up 4 cents a gallon last Friday, representing an 80 percent increase in its gasoline prices since January 1979.

Gulf, with 13 percent of the Washington market, increased its price by 3 cents a gallon Saturday without announcing it.

Sunoco, with 5 percent of the market here, went up a penny a gallon on Monday. "It's going up like that every week; it's terrible," said a spokesman.

The average price of a gallon of gasoline in Washington was $1.13 on Jan. 11, the last time the American Automobile Association checked, and analysts say that a good guess is that it is now $1.17 or more.

The Lundberg Letter, a petroleum marketing newsletter, projected that gasoline could go to $1.52 a gallon by year's end. Shell president John F. Bookout suggested last week that the number might be closer to $2, but nobody knows for sure.

The oil companies -- as Exxon did yesterday -- attribute the price increases to world crude oil price increases.

Oil dealers and service station operators say that shocked customers are trying to conserve fuel, but in some cases have already conserved all they can. t

Metropolitan's Curtis said his customers have cut heating oil purchases by 9 percent from last year's levels -- a figure adjusted so that it represents conservation and not, just warner weather.

Vic Rasheed, director of the Greater Washington-Maryland Service Station Association, said, "People have already made their cuts [in gas use], they're not using gas unnecessarily. From now on they're going to have to give up food for gas."

Joe Strain, an analyst, for Sunoco, said that heating oil customers studied in several southern states -- including parts of Virginia -- had cut consumption by 15 percent in three years -- again adjusted to represent pure conservation.

While prices soar, gasoline supplies are tighter this month than in any month since February 1975, according to Lundberg.

Strain suggested yesterday that supplies might loosen up if this winter continues to be unusually warm.

"There will be a tendency [among the big oil companies], the further we get through the winter, to begin to convert production to gasoline to get ready and have some inventories prior to the big driving season next summer," Strain said. "That will be determined by what the weather is like during the next 45 days."