Representative Nancy Pelosi of California, the House Democratic leader, arriving for a news conference last month in Washington with members of the chamber’s Financial Services Committee. She has vowed a weekly vote on the release of President Trump’s tax returns.© Al Drago/The New York Times Representative Nancy Pelosi of California, the House Democratic leader, arriving for a news conference last month in Washington with members of the chamber’s…
WASHINGTON — President Trump gives Democrats in Congress an ideal adversary, one who can set off impassioned protests and ignite their supporters with a single Twitter post.
But with the Democrats’ power severely limited by their minority status, they cannot conduct congressional investigations. The chances their bills will even get a vote are slim.

What they do possess, they hope, is the power to shame. So they have been offering up bills that stand virtually no chance of passing, like ones forcing Mr. Trump to release his tax returns, with the clear intent of putting Republicans on the record now in hopes that voters will punish them later.
Last week, Representative Nancy Pelosi of California, the House Democratic leader, vowed a weekly vote on the matter. “We’re just going to pound away,” she told colleagues in a closed-door meeting, according to an aide who was
The results so far are decidedly modest. On Friday, two House Republicans signed a Democratic letter requesting the release of the president’s tax returns, becoming the only Republicans to do so.
During the first six weeks of the Trump administration, Republicans have supported the president with a watch-what-he-does, not-what-he-says approach. It has largely been a policy of muted assent.
Democrats have eagerly filled the void with a steady trickle of measures, using arcane procedures to produce a paper trail. Some have seemingly innocuous aims, like a House bill that would create an independent, bipartisan commission to investigate foreign interference in the 2016 election. The measure has drawn just one Republican co-sponsor.
“I think we’re going at it from all angles, hoping that whether it’s the pressure or wanting to do the right thing,” Republicans will join them, said Representative Eric Swalwell of California, one of the two Democrats who introduced the bill. “Whichever comes first.”
Democrats say the tax returns provide a nexus between concerns about Mr. Trump’s personal financial conflicts of interest and his campaign’s ties to the Russian government, issues that have troubled members of the president’s own party.
“I’m for transparency,” Senator Bill Cassidy, Republican of Louisiana, told a few hundred people in a high school cafeteria in Denham Springs, La., on a recent afternoon, stopping well short of a call for legislation.

Senator Ron Wyden of Oregon, the top Democrat on the Senate Finance Committee, has led the chamber’s efforts to disclose Mr. Trump’s returns, a push he said was driven by constituent outrage.
“People just kind of gasp,” Mr. Wyden said. “I can tell you it comes up at every one of my town meetings. People think it is very much linked to the Russia issue.”
In the House, Democrats have pressed the issue on several, so far fruitless fronts. Representative Bill Pascrell Jr. of New Jersey, who wrote the letter that has drawn two Republican signatures, introduced a measure that would compel the House Committee on Ways and Means to request Mr. Trump’s tax documents, as it is empowered to do.
Mr. Pascrell used a privileged resolution, a procedure that allows even a single member to bypass leadership to bring up a measure for a vote — and that was more theatrically employed by some House conservatives in the fall in an effort to oust the head of the Internal Revenue Service over objections by their leaders. Mr. Pascrell’s measure failed in a floor vote last week along party lines.
Representative Kevin Brady, Republican of Texas and the chairman of the House Ways and Means Committee, has resisted calls to, as he described it to reporters, “rummage around in the tax returns of the president.”
A few Republicans have broken from their party on investigating connections between Mr. Trump’s campaign and Russia. Representative Darrell Issa of California, who narrowly won re-election in November and is considered vulnerable in 2018, recently called for an outside investigation. The House and Senate intelligence committees will conduct their own inquiries.
Representative Walter B. Jones of North Carolina — who, with Representative Mark Sanford of South Carolina, signed Mr. Pascrell’s letter and voted “present” on his measure — said he wanted to send a message to Republican leaders that he intended to keep “an open mind.”
Mr. Jones is also the sole Republican co-sponsor on Mr. Swalwell’s independent commission bill, calling Russian interference in the election “serious business.” (“Here you go again, Walter,” he recalled a couple of Republican colleagues saying to him.)
“If they would not let the politics drive their thinking and let the needs of the American people drive their thinking, then I think some of them would join this effort,” he said.
With control of what legislation comes to the floor in both the House and Senate, Republican leaders have easily fended off Democrats. A House measure seeking more information on Mr. Trump’s conflicts of interest and his ties to Russia was sent to the Judiciary Committee, where it was rejected along party lines last week.
The vote came at the end of the committee’s meeting, hours before Mr. Trump’s address to a joint session of Congress, when many lawmakers were having dinner. When the livestream on the committee’s website temporarily failed, Democrats streamed the meeting on their Facebook pages, a low-yield guerrilla-style tactic.
While Democrats argue that they are giving voice to concerns about the Trump administration, pressuring rivals to cast votes on a contentious subject is a standard political strategy.
Democrats are hoping to use the Republican votes against forcing disclosure of Mr. Trump’s tax returns, and on other issues, in the 2018 midterm elections.
“Now he is the president, he is their standard-bearer,” said Meredith Kelly, a spokeswoman for the Democratic Congressional Campaign Committee, the campaign arm of House Democrats. “They’ll have real votes on his agenda, from helping him hide his tax returns to helping him hide his connections to Russia.”
But Matt Gorman, a spokesman for the House Republicans’ campaign organization, the National Republican Congressional Committee, said their incumbents had established themselves with their voters, who care more about economic issues than whether a particular lawmaker pushed to see Mr. Trump’s tax returns.
“Voters won’t be swayed by these legislative tactics,” Mr. Gorman said. “They’re stunts.”
Mr. Jones said he saw the efforts to answer Americans’ lingering questions about the election as a “service” to the Trump administration, as well.
“We are an equal branch,” he said. “We have an obligation to the American people, just like the president has. And to me, I think the sooner we start dealing with this situation, the better off we are.”