[Some very nice answers by Eric, Sivaram and Piotr above. Here's my take!]
Short answer: NO !
The notion of $C^\infty$ is a mathematical aberration that was conjured up to help smooth (pun intended) discussions in real analysis.
Now, remember, you asked "Is the world $C^\infty$?". By "world" I take it to mean the physical world around us, our notions of which are based on what we can observe. A physical observable which is infinitely differentiable, would require an infinite number of measurements to determine the value of that observable in a given region.
Given that the consensus is emerging that information is the underlying substrate of the Universe (in the various forms of the holographic principle), it becomes even more urgent to reject a notion of $C^\infty$ observables.
Note how I have stressed the words "physical observables" rather than functions or mathematical entities that are used as intermediaries to compute any measured quantity. This is in harmony with Eric's statement that:
This post imported from StackExchange Physics at 2014-04-01 16:43 (UCT), posted by SE-user user346
It is also possible that smooth spacetime is some aggregate/thermodynamic approximation of discrete microstates of spacetime -- but our model of that discrete system will probably be described by the mathematics of continuous functions.