The nation's energy crunch is adding to financial pressures on local governments here and across the country as officials scratch for extra millions to pay soaring energy bills.

At the same time, gasoline tax revenues in many jurisdictions -- including the District of Columbia, Maryland and Virginia -- are plunging because of reduced gasoline supplies and sales.

In the District of Columbia, energy costs are up at least $17 million this year over last -- a 37 percent increase -- officials estimate. This comes at a time when city agencies and schools are already facing possible cuts because of a shortfall in city tax revenues and new, unanticipated costs.

The gasoline shortage alone is reducing city revenues by an average of $165,000 a month -- nearly $2 million a year -- in lost gasoline taxes, according to an analysis of U.S. Department of Energy supply statistics.

Such losses are 10 times as great or more in Maryland and Virginia, with their greater volumes of gasoline sales.

Virginia Gov. John Dalton has tried to change the flat 9 cent-a-gallon gasoline tax to a percentage figure so that soaring prices will increase tax revenues, but the state legislature has balked. Maryland Gov. Harry Hughes is studying a similar measure.

"It's very scary to have such a colossal jump in the cost of energy for the District government in spite of our vigorous conservation efforts," said D.C. energy chief Chuck Clinton. He said energy costs would be 10 to 20 percent higher than they are if the city had not begun a conservation program.

"We're doing everything from testing new (energy saving) products to managing our vehicle fleet more efficiently and buying smaller vehicles," Clinton said.

This year the city will spend $61 million on electricity, oil and natural gas to heat, cool and light its buildings. Last year it spent $44.2 million for the same utilities. In 1973 -- the time of the Arab oil embargo that first sent fuel prices skyward -- the city spent only $18.2 million for energy. f

City administrator Elijah B. Rogers last week ordered department heads to prepare for cutbacks that could freeze hiring and promotion of city employes. The pressure of rising energy costs and reduced gasoline tax revenues is at least one reason why the city is in dire financial straits.

In Virginia, state police have served notice that they need an additional $1.5 million during the first half of 1980 if they are to keep patrolling state highways with the same frequency they do now, according to Stuart Connock, the state's director of planning and budget.

Connock said he will seek authority to pay this and other increased energy costs through June by transferring $8.5 million out of a central holding account appropriated for salaries.

Since no employe could go without his pay or projected pay increases, the money would utimately come from squeezing agency budgets, Connock said.

"They'll just have to tighten up a little here and there, maybe leave a position vacant (for a short time)," he said.

In Arlington, "It's just a matter of cutting down to try to keep our head above water," according to Patrick Hyland, the county's financial analysis supervisor.

Hyland said that increased electricity rates alone will add $450,000 to the county's expenses this year, with an additional expense of perhaps $350,000 later.

At Arlington's new waste water treatment plant, an extra $1 million will be needed next fiscal year for electricity -- considered a "major, major increase," according to Hyland.

And county planners are building into next year's budget a stunning 100 percent increase in gasoline and diesel fuel costs. "We have to add half-a-million to our fleet budget," said Hyland. "We're reducing consumption by 10,000 gallons, but while we're using less we're paying more for it."

Hyland said no major program cuts are planned to help pay for all this. Again, it is a matter of squeezing and conserving -- shutting off athletic field lights when not in use, for example.

The county used to keep them burning all the time -- "something that in this day and age just doesn't make any sense any more," Hyland said.

In Fairfax County with its 160 school and other buildings and its huge fleet of police cars and school buses, county officials recently had to set aside an extra $3.2 million to cover energy-cost increases according to Edmund Castillo, the county public information officer.

In Prince George's County, county energy coordinator R. Hall Silvers found that the county government's fuel oil consumption went down 3 percent last year, but that the price of it soared as suppliers charged 50 and 100 percent more for the product.

"We are hurting, we are biting the bullet," Silvers said.