I wouldn't assume that. I can't find the terms and conditions for AANYC, but for Fly Three Fly Free, it says:

"*Free Award Ticket Taxes/Fees: The passenger is responsible for all taxes/fees, including: (i) up to $18 per round trip in local airport charges; (ii) federal excise tax of up to $3.10 per U.S. domestic flight segment (one takeoff and landing); (iii) September 11th Security Fee of $2.50 per U.S. enplanement, up to $5 per one way or $10 per round trip; (iv) federal excise tax of up to $13.80 per round trip for domestic flights beginning or ending in Alaska or Hawaii."

AA eats the PFC and segment taxes on regular award tickets.