Copies of Fairfax County's proposed new budget had hardly hit the table yesterday before most county supervisors were assailing its call for higher real estate taxes.

The proposal that united the supervisors calls for taxes on the average Fairfax home to rise by 14 percent next year, largely to fund big increases in spending on schools, energy and Metro.

The supervisors, who must approve both the budget and county tax rates, were openly hostile to acting County Executive J. Hamilton Lambert's tax proposals. "Under no circumstances am I going to support it," announced Board Chairman John F. Herrity. "We will have to cut spending."

Herrity, a conservative Republican, was not alone. Democrat Martha V. Pennino, the board's vice chairman, cited the impact of soaring property assessment on homeowners as she denounced Lambert's call for a 4-cent increase in the $1.54 real estate tax rate.

"If we have an increase in revenues from assessments, why do we need a 4-cent increase?" Pennino asked. "We'll have to cut the budget someplace."

"I could never support an increase in the tax rate," agreed Supervisor Nancy F. Falck, who said rapidly rising assessments are hitting many homeowners in her Dranesville district.

The operating budget that Lambert sent the Board of Supervisors yesterday calls for spending of $446.5 million, up 16 percent from the $384.4 million budgeted for the current year ending June 30.

Homeowners, who foot the costs for most of Fairfax's budget, would pay for the increase under Lambert's plan through the 4-cent rise in real estate taxes and through an 11 percent rise in assessments.

Taxes for an average home, valued at $69,060 in 1979, would jump from $1,063 last year to $1,211 under the impact of Lambert's proposal.

His proposal calls for the first increase in the county's real estate tax rate since 1976.

Newly installed supervisor Thomas M. Davis III (R-Mason) spoke for many supervisors when he warned county officials "will have a lot of convincing to do before I'll support" the increase.

Lambert's budget adopted a $37.3 million increase in school spending proposed by the county school board. Although Lambert is not empowered to cut the education budget, the supervisors are and several have stated they have misgivings about the education package.

In projecting a 16 percent increase in the cost of county government for fiscal 1981, Lambert also forecast a $9.4 million increase in energy costs and $5.7 million increase in Metro transit.

The county does not have much flexibility in cutting energy costs, but many of the supervisors are talking about adopting a much tougher position on Metro bus operations, which account for about two-thirds of the mass transit costs.

Supervisor Marie B. Travesky said "we'll have to look at the kind of bus service we have. Do we need midday service? Can we end more bus routes at Metro [subway stations]?"

The Lambert budget calls for an 8.5 percent cost-of-living increases for county workers next year.

Lambert also proposed adding 103 employes to the payroll -- most of them fire and rescue services (40) and police (28). He said there were departmental requests for more than 500 new positions.

The county executive said his budget "does not contain major new funding proposals or (general) program expansions . . . Rather, the small increases that are included are necessary to continue most programs at existing service levels."

While the budget calls for a small tax increase the county -- unlike the District -- is in good financial health.

In fact, the budget portrays a county that is prospering to such a degree that by fiscal 1984, it should have a $4.2 million surplus, and a $7.5 million surplus the following year.

Per capita spending is down and so is the ratio of debt to revenues -- the key factor in the Fairfax's Triple A bond rating, (a distinction enjoyed by only nine of the county's 3,300 counties.

"The county is in a superb financial posture," said Deputy County executive James P. McDonald, noting that it can sell general obligation bonds at a 6 to 6 1/2 percent interest rate. With the county likely to sell nearly $200 million worth of bonds over the next five years, the low rate would result in millions of dollars saved in interest costs, he said.