WASHINGTON—The Internal Revenue Service is making changes to its partnership with private companies that provide free tax preparation to millions of filers, following intense criticism of the program earlier this year.

The public-private partnership, called the Free File Alliance, was designed to provide free tax prep by a dozen private companies, such as H&R Block Inc. and Intuit Inc., the maker of TurboTax, to about 100 million filers earning about $69,000 or less.

In...