WASHINGTON—Rep. Richard Neal issued subpoenas Friday to the Treasury secretary and IRS chief seeking six years of President Trump’s tax returns and audit records, taking the next formal step in the dispute between House Democrats and the Treasury Department.

Democrats don’t expect Treasury Secretary Steven Mnuchin or Charles Rettig, commissioner of the Internal Revenue Service, to comply with the subpoenas, following Mr. Mnuchin’s rejection of a previous written request from Mr. Neal and following the administration’s general...