One of my fig trees is putting out two little suckers, each one with only one set of leaves so far. I've heard that a new tree can be grown from these, I've also heard that when left on, can be the best producing part of the tree. Any input? If I were to remove the sucker, what would be the technique and what time of the season should I do it? Thanks for any help.

If you leave them you will get a bushy plant. Pinch them back and the main stem will get all the nutrients. I never get a fig that grow suckers. :(
I don't believe pinching at the wrong time(if such a thing exist) will cause harm. 

I just used my shovel to chop & pry 6-Neveralla suckers from the parent tree! Growing low, They wouldn't get any sun so it was best to dislodge them and try to get some of the small roots with them. 
I hope to propagate. Anyhow, they are all in soil and covered with a plastic bag.

If you want to take them off cover the bottom with soil so you get roots then plant them or sell them.  I leave mine on.

Two Saturdays ago I took a big sucker from the base of my 3yr.old Kadota...the trick (as mentionrd above) is to dig down deep enough at the base of the tree to see where the sucker attaches to the tree, and to see if you can take enough roots with it so it survives the surgery and the transplant...the Kadota sucker was thick and strong and had a nice little root system of it's own, so the transplant to a 1gal. pot was successful. (no wilting or leaf drop...it's just as shiny and strong as when it was still attached to the mother tree.)......good luck with your suckers(it's a "new" tree for free).   :)  :)

I have in past pulled a few suckers with roots , airlayered a few over the years by staking them down into the mix etc they all grow very fast.

my VdB and White Greek like to put on suckers. with VdB, i used to let them grow, but it seems they take the energy away from the main tree. the suckers grow much faster than the main tree. now, any new suckers coming out gets pinched off.

Do an air-layer  using a foam cup, 4-8 weeks you have plants