I’m a foreign student studying in the U.S. Does the requirement to have health coverage apply to me?

In general, yes, through 2018.  Starting in 2019, there will be no tax penalty for not having health insurance.

For 2018, people who might otherwise owe a tax penalty can claim an exemption from the penalty directly on their tax return if they experienced a hardship that prevented them from getting coverage.  No documentation is required to be submitted with the tax return, though taxpayers should retain any documentation of the hardship for their own records