What that would mean is that most House members wouldn't have to return to the Capitol for an actual roll call vote on the issue. They would seek "unanimous consent" on the House floor to okay the bill.
Keep in mind, unanimous consent is a normal way of operating in the Senate and House. They approve resolutions, bills and other issues by unanimous consent all the time.
It only takes one member to object, however, and blow up the bill (thus, the "consent" is not "unanimous"). But without an objection, they only need a skeleton crew on the floor to approve the legislation.
This gives Boehner at least several outs even if his side doesn't like the language of the Senate bill.
But therein lies the problem for Boehner. Democrats know he can't get clearance on his side to do that (which is the reason for the holdup).
So Democrats are sort of sticking this in Boehner's eye, saying "Look, we have our side organized, can't you get your side together?"
Furthermore, Boehner could pass the Senate bill with about 80 GOP votes and the rest coming from Democrats, and he would face a complete insurrection from his side, questioning his fitness to be speaker.
The following is the body of the letter to Boehner:
"We are disappointed that during yesterday morning’s House session, the speaker pro-tempore refused to allow the consideration of the bipartisan Senate-passed bill to extend the payroll tax cut, unemployment insurance and the SGR Medicare physician fix.
"Passage of this bill is essential prior to January 1, and is the only way to ensure that tens of millions of Americans avoid losing tax cuts and other essential benefits as we continue to work on a longer-term extension of these critical policies.
"Yesterday, 180 Democrats introduced HR 3743 which is identical to the Senate-passed bill. The House has three chances before January 1 -- this Friday, next Tuesday, and next Friday -- during which this bill could be passed by unanimous consent.
"Democrats will provide that support to provide the American people a firm guarantee that these essential benefits will not terminate unnecessarily on December 31. We urge you to direct whoever is serving as the presiding officer on each of these upcoming House sessions to recognize a Democratic member to offer a motion to pass the short term extension without further delay. There is no reason why such an extension cannot be on the president’s desk within a few hours of House action given the overwhelming support the Senate has already demonstrated.
"We owe it to 160 million affected Americans to pass this consensus, bipartisan bill before December 31, and Democrats offer our support as soon as you will allow the bill to be considered."