The eaves were dripping, the curbs were dripping, and rivulets of water were flowing out from under ancient mounds of ice into the nearest gutters yesterday as the warmest air in three weeks flowed into the Washington area.

The respite from the bitter weather is expected to last at least through today, with temperatures reaching or exceeding yesterday's high of 43 degrees, according to the National Weather Service.

However, forecaster John Quadros said the weather is expected to turn colder again on Thursday, with Friday's temperatures returning to the freezing range.

After weeks of watching their streets and driveways freeze solid, after long mornings spent trying to start cars that didn't want to go, area residents treated yesterday's normal winter temperatures like a breath from the tropics.

Coats were unbuttoned, gloves and hats were left inside at lunchtime, and pedestrians had to dodge growing puddles and the muddy spray sent out by car wheels.

But while the city and suburbs coped with the slush, mud and water left by the brief thaw, local governments were still totaling up the stack of bills left by the earlier frigid weather and worrying about people venturing out onto the softening ice that still coats area waterways.

At a Montgomery County Council meeting yesterday, John M. Brusnighan, acting general manager of the Washington Suburban Sanitary Commission, said the rash of water main breaks caused by the cold has meant a heavy financial burden for the WSSC, which will probably be passed along to water users in Montgomery and Prince George's counties.

The total cost to users will be "at least" $250,000, Brusnighan told a reporter after the meeting. He told the Council that he will eventually have to ask its permission to dip into the agency's $500,000 emergency fund to cover agency expenses incurred because of the winter storms.

The agency announced Monday that, "for the remainder of the emergency" the WSSC is putting maintenance crews and supervisors on six-day-a-week, 12-hour-a-day schedules, instead of their normal 40-hour weeks.

Last week the agency reported that in the last six weeks there have been more than 220 water main breaks - more than half the average number for a year - in its 3,300-mile water supply network.

While yesterday's thaw offered a change for WSSC crews, the thick ice left covering the roads, rivers and bays of the area, still proved to be treacherous.

A District of Columbia construction employee was critically injured yesterday when he slipped on the ice as he was directing traffic at the Lorton, Va., landfill, and fell under the wheels of a truck.

Benjamin Murrel, 30, of 1538 Butler St. SE, was reported in critical condition at Potomac Hospital with a severed left leg and a crushed right leg and pelvis. The District government leases the landfill where he was working from Fairfax County.

On the Chesapeake Bay yesterday, a tanker filled with almost 6 million gallons of gasoline ran aground as it approached the ice-covered waters of Baltimore Harbor. The Coast Guard said no gasoline has spilled or leaked.

A Coast Guard spokesman added that, while the cause of the grounding of the tanker Overseas Alice is "still under investigation," some of the navigational buoys in the area have been submerged or pushed out of line by the ice.

The temporary thaw in this area was generated by a high pressure system now sitting over the Gulf of Mexico. It pushed warmer air as far north as New England, helping to melt the three to five-inch snowfall that preceded it. Much of the rest of the country continued to struggle with the energy crisis caused by the cold.

Georgia officials said yesterday that a worsening natural gas shortage has forced the layoffs of about 75,000 workers, at a cost of $30 million to those idled.

A total of nearly 250,000 workers in Tennessee, Pennsylvania, West Virginia, Florida and Mississippi have been idled by power and gas cutbacks and other problems.

In Ohio, as part of an effort to keep about one million residential customers warm through the winter, Columbia Gas of Ohio said it would have to cut off gas to 15,000 business customers - a move that could lead to heavy layoffs.

In Cleveland, the shortage of natural gas prompted Mayor Ralph Perk to initiate a search for fuel on city land. If exploration showed that there was potential for finding natural gas somewhere on city property, Perk announced, drilling would take place.

President Carter has said that he will submit energy legislation to Congress today to facilitate the flow of natural gas in interstate pipelines - a flow now under heavy federal regulation.

In other areas the legacy of the persistently low temperatures has been empty water pipes. In the Alabama mountain village of Gaylesville the townspeople have drained their well by pumping too much water in an effort to keep their pipes from freezing.

Closer to Washington, in the town of Bloomington (pop. about 800) in western Maryland, residents last week had to dig down to water pipes and light bonfires on frozen sections to get water flowing back into their taps.

While the fast-rising temperatures left roads clear enough for a smooth morning rush hour yesterday, the thaw also caused it own problems: potholes.

District engineers reported yesterday that 21 trucks - the same vehicles used for plowing the streets - were out on the roads filling the deepening holes. But there was a sense of futility about the job. "It'll freeze again tonight and there'll be more potholes tomorrow," said Stanley Ather, a street maintenance engineer.