Join Date: Feb 2000
Location: San Francisco (Marin County), CA
Programs: UA Global Services, AA Lifetime ExecPlat, LH HON Circle, SQ PPS Solitaire, Hyatt Courtesy Card
Posts: 3,002
No, you do not. There is no FF program in the industry which allows you to earn miles on free tickets.
I guess the only thing even close I can think of is, at a Hyatt you can get a stay credit for a free night if you purchase some incidentals at the hotel. No such luck with any airline.