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The new legislation, which President Donald Trump signed into law last week, caps at $10,000 the amount of state and local taxes that filers can deduct from their federal tax bill. That means those whose tax bills regularly exceed that amount could benefit by paying more tax in 2017, when the deduction has no limit.

Municipal offices in a number of states saw a busy post-Christmas rush on Tuesday as taxpayers calculated the effects of the new law. In Fairfax County, Va., where property values have risen sharply in recent years, hundreds of people lined up at the government center to prepay. Across the Potomac River, in Montgomery County, Md., the county council held a special session Tuesday morning to pass legislation allowing residents to prepay their taxes.

The new tax law “is a middle-class tax hike for us, and we’re trying to postpone at least some of that for at least one more year,” said Montgomery County Council President Hans Riemer.

In Massachusetts, local treasurers welcomed the sudden surge in tax receipts. “It’s been insane here,” said James McAuliffe, the town treasurer in the Boston suburb of Milton, adding that he went to the bank Tuesday morning to deposit early property-tax receipts and would likely have to go again in the afternoon.

“Thank you, Mr. Trump, for solving my cash-flow issues,” said Mr. McAuliffe, who estimated that about half the residents of his municipality would hit the new $10,000 cap. “It’s become a very expensive town.”

The $10,000 limit also covers state and local income and sales taxes, but lawmakers drafting the bill barred people from prepaying those other levies. “It left it up to the localities whether or not they would allow you to prepay” property taxes, said Nicole Kaeding, an economist at the Tax Foundation.

Some officials urged caution on prepaying property taxes, given that the new law didn’t address the issue. In New Jersey, most towns aren’t encouraging residents to prepay amid questions about whether the Internal Revenue Service would allow tax filers to take the deduction early, said Michael J. Darcy, executive director of the New Jersey State League of Municipalities.

The IRS didn’t return a request for comment.

Officials in Westchester County, N.Y., just north of New York City, said the county wouldn’t be able to calculate final tax obligations for each of its municipalities before year’s end. Connecticut, meantime, isn’t allowing prepayment of property taxes.

Nationwide, about a quarter of taxpayers deducted their real estate taxes in 2015, the latest year available, shaving roughly $5,000 from the average tax bill, according to Tax Policy Center data. At least one taxpayer in three deducted those taxes in Virginia, Maryland, New Jersey and Massachusetts.

“This matters most in areas that have higher levels of state and local income taxes,” Ms. Kaeding said. “Think the D.C. area, New York, California, Connecticut, New Jersey.”

The drafters of the tax bill say the unlimited deduction simply cost the federal government too much money and encouraged big spending by state and local governments.

Congress’s Joint Committee on Taxation estimates the real estate tax deduction resulted in about $33 billion less revenue for the federal government in 2016. The $10,000 cap was intended to raise new revenue and offset some of the effect of the legislation’s tax cuts.

Many of those worried about less of a property-tax deduction next year could still see a lower overall tax bill, because the GOP tax package also lowers federal income-tax rates, said Kim Rueben, a senior fellow at the Tax Policy Center.

As property values have appreciated, homeownership has often been the best-performing family asset, especially in expensive coastal states. Yet the rush to prepay property taxes on Tuesday reflects how the increased tax liabilities for some homeowners may erode the so-called wealth effect of rising property values.

Home prices nationwide were up 6.2% in October from the previous year, according to data from the S&P CoreLogic Case-Shiller National Home Price Index released Tuesday, up from a 6.1% year-over-year increase in September. The increase in home values has been accelerating for 16 consecutive months.

Weston, Mass., has some of the highest property taxes in the state, averaging around $18,000, said town treasurer Peter Forcellese. Local officials are getting 60 to 70 calls a day from people with questions about prepaying.

On Friday, several hundred residents came in to prepay, and a steady stream is expected all week, he said. “There is definitely a lot of activity,” Mr. Forcellese said. “There are a lot of folks looking to get the tax payment in and get inasmuch as they can.”

Sacramento County, Calif., is allowing residents to pay by year-end a property-tax installment due next April. Some would pay even more if they could, but the county isn’t yet accepting payments for tax bills further out, county spokeswoman Kim Nava said.

Santa Clara County, in high-cost Silicon Valley, said collections for April 2018 installments already total $422 million, compared with $303 million at this time last year.

Bethesda, Md., resident Leroy Walters said he planned to prepay his taxes because he typically gets above the new $10,000 cap in state and local income and property taxes. “We would even be willing to borrow against a home-equity line of credit in order to prepay,” said Mr. Walters, a retired Georgetown University professor.

In Virginia, Fairfax County tax officials told residents they could prepay their estimated 2018 tax but couldn’t guarantee that payments received after Tuesday would be processed by year’s end. That announcement sent a rush of people to the tax collector’s office.

“What we’re seeing today is unprecedented,” said Scott Sizemore, director of revenue collection for Fairfax County. “People seem to be taken aback by the line, but they seem to have anticipated there would be a potential wait.”

In New York state, Gov. Andrew Cuomo  on Friday signed an executive order setting in motion the process for paying 2018 property taxes in advance and suspending laws that limit partial tax payments. Local officials who administer that process are scrambling to make accommodations.

In Livingston County, in New York’s upstate Finger Lakes region, the county is offering to answer residents’ questions about how much they will owe in 2018 so that they can mail in checks beginning Friday, even if their town tax collectors’ office is closed.

“With the holidays this creates a bit of a challenge,” said County Administrator Ian Coyle.

To prepay or not?

—Kristina Peterson, Heather Gillers, Chris Gordon and Joseph De Avila contributed to this article.