I went through and counted every vote for an actual FF game (discounted FFL because it's really the SaGa series, discounted Crono series even though I love Trigger, but kept things like Mystic Quest - for real) where the voter listed at least three entries. For people who listed more than three, I only took the top three. I did not otherwise take into account any aspect of order, since some people may not have been listing in order of preference. I probably also miscounted, so consider these values to have a ±2 margin of error (to pull a figure out of my rear end). Also, for one person they mentioned FF3 but I got the sense they meant FFVI, yet others apparently specifically meant FFIII (really?), so I only made one judgment call; everything else I took literally.
In order of popularity:
VI - 28 VII - 25 IX - 22 X - 19 VIII - 17 IV - 17 XII - 15 Tactics - 12 XIII - 7 V - 5 XIII2 - 4 XI - 4 III - 4 I - 4 X2 - 1 Crisis Core - 1 Mystic Quest - 1 II - 0
I was gratified to see VI was definitively the favorite. It is definitely one of the very best. I was very surprised by how high IX was (I liked it, but I didn't love it, and I didn't realize so many others ranked it highly). I expected Tactics to rank slightly higher, but it did respectably in any case.
EDITED to include SandOnMyBoot's post (immediately preceding post, ninja'd this one).
More data for you
http://www.gamefaqs.com/boards/681990-lightning-returns-final-fantasy-xiii/63924671?results=1#1 --- I disapprove of what you say, but I will defend to the death your right to say it