Mayor Marion Barry, facing a much larger potential city budget deficit this year than he had expected, promised yesterday that his administration would do a better job of handling the budget problem than it did when confronted with a similar crisis in 1980.

Barry, speaking at his monthly press conference, said that while he will not propose tax increases for D.C. residents who already face the area's heaviest tax burden, he would not rule out signing tax increase proposals that might come to his desk from the City Council.

He said there will be no cutbacks in education, corrections, the police force or the fire department, but indicated that all other city agencies and functions are potential targets. He reiterated a promise that there would be no layoffs of city workers during the next four years.

Barry's disclosure over the weekend of a potential $110 million deficit -- just over 5 percent of the District's total budget of about $2 billion--comes less than a month after his successful reelection campaign in which he took credit for having resolved the earlier budget crisis and a week after city officials said they expected a shortfall in city revenues of about $20 million.

"The point I want to make," Barry told reporters, "is that I know what is going on. Unlike three years ago, when we discovered all these problems and didn't know what they were, we're on top of things. We know exactly the magnitude of the problem; we have some tentative solutions we are looking at."

Barry added, "But we're not going to let the budget be the tail that wagged the dog" -- a reference to his own description of how his administration became obsessed with the budget in the previous crisis, letting some other governmental functions go unattended while he and his top aides did chores like counting the city's supplies.

To begin saving money, Barry said, the city has already cut back on hiring and last Friday issued new reduced budget ceilings for city departments. But Barry declined to reveal the extent of the budget reductions. The mayor said that some city agencies will be merged as an economy measure, and that the city is likely to increase many of its 800 user fees to bring in more cash.

In addition, Barry said, the city probably will not pay an $18 million increase in its Metro transit subsidy next year.

Barry said that while he believes the city needs more revenues to function, he is philosophically opposed to proposing tax increases.

"We can't continue to run businesses out of the city because of our taxing policies, or run middle-income or moderate-income people out of the city, therefore narrowing the tax base even smaller and leaving the dependent population here," he said.

Philip Dearborn, vice president of the Greater Washington Research Center, which released a report Monday accurately projecting a $109 million deficit for the city this year and predicting further deficits the next four years, said yesterday, "I don't see any likely solution that would not include tax increases. It's going to be awfully difficult to save $110 million out of a budget that already has two months gone."

Barry said he would release a plan for dealing with the problem in January, and said the approach would entail reductions in both the current fiscal 1983 and forthcoming fiscal 1984 budgets.

Barry said he has not yet adopted a position on any tax increases the City Council might initiate. "I haven't seen anyone in the council introducing any legislation to increase any taxes," Barry said. "When they introduce it, I'll have a position on it."

In a letter to City Council Chairman Arrington Dixon, disclosing the potential deficit, Barry said revenues would fall short by about $51 million and city agencies were overspending at a rate of $58.9 million.

Most of the overspending, $45 million, is in the Department of Human Services. Most of that money, about $26 million, is in administrative costs, such as unexpected increases in building rents and higher personnel costs. Overspending is also occurring in programs such as Medicaid ($12.3 million) and foster care ($7 million).

The remainder of the overspending, about $13 million, is occurring primarily in the prison system and low-cost housing programs.