Donations are pouring into Monroe County Hospital in Forsyth, Ga., offering a lifeline that is expected to close a $1 million operating deficit and help preserve local medical care and jobs.

The lifeline may not last, however, because it could be collateral damage from a coming Treasury Department regulation aimed at curbing efforts by New York, New Jersey and other high-tax states to work around deduction caps in the new U.S. tax law.

In...