According to this article from the Austin American-Statesman, some refineries may shut down in the event of a strike by refinery workers, while others may look to non-union workers. Fortunately, a strike was averted on Saturday when both sides agreed to continue contract negotiations.
[Update. From the Houston Chronicle: It appears that the parties reached an agreement giving refinery and chemical plant workers a 3 percent raise each year for the next three years along with a $2,500 contract ratification bonus.]