Homeowners across the country are challenging their property tax bills in droves as the value of their homes drop, threatening local governments with another big drain on their budgets.

The requests are coming in record numbers, from owners of $10 million estates and one-bedroom bungalows, from residents of the high-tax enclaves surrounding New York City, and from taxpayers in the Rust Belt and states like Arizona, Florida and California, where whole towns have been devastated by the housing bust.

“It’s worthy of a Dickens story,” said Gus Kramer, the assessor in Contra Costa County, Calif., outside San Francisco. “These people are desperate. They know their home’s gone down in value. They’ve watched their neighborhoods being boarded up. They literally stand in there and say: ‘When can I have my refund check? I need to feed my family. I need to pay my electric bill.’ ”

The tax appeals and reassessments present a new budget nightmare for governments. In a survey conducted by the National Association of Counties, 76 percent of large counties said that falling property tax revenue was significantly affecting their budgets, said Jacqueline Byers, the association’s research director.