From the American Petroleum Institute:
Contrary to what some in politics and the media have said, the oil and natural gas industry currently enjoys no unique tax credits or deductions. Since its inception, the U.S. tax code has allowed corporate tax payers the ability to recover costs and to be taxed only on net income. These cost recovery mechanisms, also known in policy circles as “tax expenditures”, should in no way be confused with “subsidies”, i.e., direct government spending.
Read the details here.
In addition to the misunderstanding that oil companies somehow receive direct subsidy payments from the government like the payments some farmers receive (sometimes for NOT growing certain crops), there is also the misunderstanding that oil and gas companies are "not paying their fair share" of income taxes. The assumption here must be that other companies or U.S. industries are paying "their fair share" and oil companies "get off easy."
The chart above is based on data from the API and shows that oil companies face a much higher income tax burden as a share of before-tax earnings (41.1%) compared to other S&P Industrial corporations. Keep in mind that oil companies pay a higher effective tax rate than even the highest corporate marginal income tax rate of 35% because they are also paying state income taxes and foreign income taxes. And there are also royalties, bonuses bids, excise taxes and other transfers to governments that aren't even included in the 41.1% tax expense rate.