Even without the support of the DUP the government can continue to govern, relying on abstentions, small scale deals, etc to pass votes. Such a government would lack the authority of one with a majority as they would be unable to pursue all the actions they want to. However, a minority government could still function.
Minority governments have existed in the past and have survived for several years. For example, John Major led a minority government for a few years prior to the '97 general election.
There are three basic paths for the current government to collapse:
1) The government loses a specific vote of no confidence. The required wording is set in the Fixed Term Parliaments act (FTPA). Note that if the DUP abstained in such a vote, the government would likely have enough votes to survive.
Following a successful vote of no confidence, there would be a 14 day period for anyone to attempt to form a government that could win a confidence motion. If no one is able to do so, a general election would be called.
2) Theresa May either resigns or is removed by her party. This would result in the end of the May government. Once May has visited the Palace to offer her resignation to the Queen, the Queen will ask someone new to form a government, by convention following the suggestion of the outgoing PM (i.e. May). It is highly likely May would suggest another Conservative MP, either as a caretaker or a permanent replacement (depending on internal Tory party politics).
3) 66% of MPs support a vote to call an early general election. This is the mechanism by which the 2017 election was called.
Note that prior to the passing of the FTPA a loss in various types of votes, including those on the Queen's speech or a budget would have by convention resulted in the calling of a general election. With the FTPA this is no longer the case - in effect the FTPA makes it easier for minority governments to survive.
