Hard say. The antenna-specific sections of the electrical code, sec. 810, are really for outside antennas. In the absence of specific guidelines, one is left with these sections of sec. 250-2, "General Requirements for Grounding and Bonding":
(b) Grounding of Electrical Equipment. Conductive materials enclosing electrical conductors or equipment, or forming part of such equipment, shall be connected to earth so as to limit the voltage to ground on these materials. Where the electrical system is required to be grounded, these materials shall be connected together and to the supply system grounded conductor as specified by this article. Where the electrical system is not solidly grounded, these materials shall be connected together in a manner that establishes an effective path for fault current.
(c) Bonding of Electrically Conductive Materials and Other Equipment. Electrically conductive materials, such as metal water piping, metal gas piping, and structural steel members, that are likely to become energized shall be bonded as specified by this article to the supply system grounded conductor or, in the case of an ungrounded electrical system, to the electrical system grounded equipment, in a manner that establishes an effective path for fault current.
So, do you have to ground the mast? Probably not, although it's up to the local inspector? Is it a good idea? Yea, especially if you have alot of electrical wiring running around the attic as well.