The DUP have threatened to “vote down the budget” if their requirements for Brexit are not met.
But what does this mean in practice? Will the Government have trouble passing a budget in this event?
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The convention in the UK House of Commons is that the budget vote also works as a confidence vote in the government. So losing a vote on the budget is taken as a vote of no confidence in the government (as is the vote on the Queen's Speech). The last time the government lost a budget vote was in 1885, when Gladstone's government lost 252-264 and the government resigned the following day.
Much of UK politics is governed by convention rather than law. The convention was that after losing a confidence vote the government had to call a general election. But there was no law requiring them to do so.
The situation changed in 2011 with the Fixed Term Parliament Act (FTPA) which legally requires the government that loses a no-confidence vote to win a positive confidence vote within 14 days or be forced to call a general election. The Budget and Queen's Speech votes do not count as no-confidence votes for this purpose (thanks @MikeScott for pointing that out in the comments).
The DUP could first vote against the budget and later vote for the government in a confidence / no confidence motion, where before 2011, the budget vote itself would trigger a general election. And the government wouldn't be (legally) required to hold a confidence vote if they lost a Budget vote. They weren't legally required to previous to the FTPA either but in practical political terms they were forced to; but the FTPA setting out the confidence / no confidence terms more explicitly may give them strong enough grounds to resist.
Whether Mrs May could survive as prime minister following losing a Budget vote, and whether the government could plausibly continue in the event of Mrs May resigning, are questions that are very debateable, regardless of what the provisions of the FTPA allow.
The budget is the annual finance act of parliament. The government needs to raise money by tax and the budget gives the government parliamentary approval for this. Without a budget being passed the government cannot function.
Normally a government party either has a majority in the House of Commons, or is in a coalition with another party which gives it a majority. These parties then instruct their MPs to vote for the Budget, which allows the government to operate.
However, the current government does not have a majority. It depends on an agreement with the DUP. Previously the DUP has agreed to vote with the government on "Finance and Confidence" motions. Thus the government is able to proceed with its business. The DUP are threatening to break this agreement if the "deal" with the EU breaks their "red lines". Roughly these are that exactly the same customs arrangements that apply to Great Britain (England, Scotland, and Wales) should also apply to Northern Ireland.
There are currently 316 Conservative MPs, and 315 MPs in opposition parties that are likely to vote against the government in a budget vote (excluding Sinn Fein, who do not vote in the UK parliament). Given the partisan nature of the Commons, it is unlikely for opposition MPs to vote with the Government on a finance bill. Thus the position of the DUP’s 10 MPs is critical. If the DUP vote for the bill it will pass, and if they vote against it will fail.
If the budget is not passed, then the government cannot function, and a confidence vote is almost certain to follow. (The fixed term parliments act means that an election not automatically called) If the budget is not passed but government nevertheless passes a confidence vote, we are in uncharted territory. The EU may decide to change the agreement, or it may result in UK leaving with no agreed deal on the future relationship.
If the government doesn't pass a confidence vote then the likelihood is new elections. The result of those elections would determine what would happen next.
This may all be sabre rattling. Robert Peston thinks that the deal that officials have negotiated is likely to be acceptable to the DUP, but they just want to remind the PM that they can't be taken for granted.
Just to add some context to the other answers, the current Conservative government lost its majority at the last election. In other words it doesn't have more MPs than the rest of the parties combined, so if all other parties vote against the budget it won't pass.
This is a dire situation and makes it extremely difficult to govern, so the Tories did a deal with the DUP. In exchange for £1.5 billion of extra spending in Northern Ireland and an effective veto over the government's Brexit policy the DUP agreed to support the Tories by voting with them, giving the two combined a majority. It's a kind of informal coalition but without DUP ministers in government roles.
As such the DUP can choose to walk away at any time. Although that would presumably cancel any of the £1.5 billion not already spent in Northern Ireland, as it was effectively a bribe, it would also allow them to block an Brexit deal that they don't like.