Exxon Mobil Corp. and New York’s attorney general are headed for a showdown this week over accusations the company deceived investors, a rare trial over how the oil industry accounts for the impact of climate change.

The trial, which begins Tuesday in state court in Manhattan, is the culmination of a sprawling investigation into Exxon and its accounting practices that spanned four years and three New York attorneys general. It is expected to include as a witness former Secretary of State Rex Tillerson, who was Exxon’s chief...