It is not clear how much leverage they have. But they might have some:
Pelosi attacks Obama-GOP tax plan as House Democrats signal fight: Russell Berman: House Majority Leader Steny Hoyer said Tuesday there was “no consensus or agreement reached by House leaders” on the deal Obama negotiated with the GOP, while Speaker Nancy Pelosi (D-Calif.) criticized GOP provisions in the agreement. In a post on Twitter, Pelosi said the GOP provisions in the tax proposal would add to the deficit and help the rich without creating jobs. The GOP provisions “help only wealthiest 3%, don't create jobs & add tens of billions to deficit,” the Pelosi tweet said. The Speaker expanded on her criticism of the Republican proposals in a statement that notably withheld any commitment of support.
We will continue discussions with the president and our caucus in the days ahead. Democratic priorities remain clear: to provide a tax cut for working families, to promote policies that produce jobs and economic growth, and to assist millions of our fellow Americans who have lost their jobs through no fault of their own.
Hoyer (D-Md.) at his weekly press conference reiterated Democratic opposition to extending tax cuts for the wealthy and said House leaders would be discussing the deal with the Democratic Caucus “over the next few days.” “We'd like the Senate to move first on this issue,” Hoyer told reporters. He said House Democratic leaders gave no commitment to the president:
We had a long meeting yesterday with the president, and there was at that point in time consensus or agreement reached by the House leadership. There was a discussion. It took some time, but there was no agreement reached.
The full caucus plans to meet Tuesday evening to consider the proposal. Hoyer stopped short of saying House Democrats would demand changes to the deal, but he warned: “It’s got to pass both houses.” Earlier on Tuesday, Rep. Chris Van Hollen (D-Md.), assistant to the Speaker, also said House Democrats had not signed off on the deal. He said he would meet with the caucus Tuesday to discuss it...
The debt-limit increase and two-year periods for all of the bill's provisions are obvious things to hold out for.