I am skeptical that butter from yogurt is a thing.
When yogurt is made the milk proteins denature and form a mesh that traps all the large molecules in the milk. Water, sugar, and some small molecules can come out but the fat never does- it's huge and tightly bound up in the gel.
Even when yogurt is blended up the whey will separate out but the fat never does. I have a hard time believe that extricating the fat from that protein mesh will be easier than just letting it rise to the surface in milk.
Perhaps thoroughly cooking the yogurt would melt out the fat but you would still lose the rest of the milk solids and would have ghee.
With purchasable yogurt it would be even less feasible because it is often made from low-fat milk to begin with.
EDIT-
Searching online I was able to find Indian recipes for making butter from yogurt:
http://www.youtube.com/watch?v=_vK8hW_oSu0
In this case they made yogurt from heavy cream. The goal was a cultured butter- not easier butter.
Creating the right environment for bacterial fermentation
Yogurt is made using a few different bacteria which excel at digesting the lactose in milk at warm temperatures. Each of those factors (culture, temperature, composition of milk,) is important in creating an ideal environment for bacterial fermentation.
You would not try fermenting yogurt using yeast or a bacterial culture different from yogurt starter, and you wouldn't expect the yogurt to ferment if you held it your refrigerator instead of a warm thermos. The composition of the milk is equally important. If you change a major component of the milk, such as by adding sugar, some fermentation may happen but the result will not be what you expect.
"Live active culture" vs. actively culturing
There is a difference between bacteria that is "alive" or "active" and bacteria that is currently doing something.
Imagine dry baking yeast in a canister. The yeast is alive, but inert. It's not in an environment where it can do anything. Add moisture and carbohydrates for it to act on, and it goes nuts. Put that dough in the refrigerator, and it slows way down. Until you cook the yeast it doesn't "die" or "deactivate" it just has various states of activity depending on its environment.
The yogurt culture is alive when you add it to the milk, alive when you're done incubating it, alive when you add sugar, and alive when you eat it, But it can't turn milk into yogurt unless you give it a very precise environment -- the right culture, the right temperature, the right environment in the milk, including the right amount of sugar.
Best Answer
Yogurt is a protein mesh that traps the rest of the milk components. Many of the trapped components are water soluble- in particular un-denatured albumin, residual lactose, lactic acid, and riboflavin. The water and these water soluble components are the whey.
Draining off the whey makes the yogurt thicker, and sweeter as some of the acid washes away.
Stir the yogurt well to break up as much of the protein mesh as possible and free the trapped whey.
Then do as Elendil suggests and hang the yogurt in a cloth to drain for a couple hours. I use sturdy mesh cloth from the remnants pile at the fabric store. Cheesecloth is too fragile to be used very many times. How many times can cheesecloth be reused?