What force causes entropy to increase? What force causes entropy to increase?  
I realize that the second law of thermodynamics requires the entropy of a system to increase over time. For example, gas stored in a canister, if opened inside a vacuum chamber, will expand to fill the chamber. 
But I’m not clear on what force, exactly, is acting upon the molecules of gas that causes them to fly out of the opened canister and fill the chamber.
Just looking for a concise explanation as to what is going on at the fundamental level, since obviously, the second law of thermodynamics is not a force and therefore does not cause anything to happen.
 A: All forces increase entropy if they result in change. The only systems not increasing in entropy are either perfectly static, or already at maximum entropy.
In the case of gas expanding to fill a vacuum, the molecules in the gas are moving around chaotically inside the container. When the seal breaks, that random motion allows molecules to cross the boundary as a matter of course. Given that all the molecules start on one side of the boundary, the next event is certain: a molecule crosses the boundary into the "vacuum" side. Molecules can cross in both directions, but the "into the vacuum side" possibility remains more probable until the molecules are evenly distributed in the new space. At that point, movement in both directions is equally likely, and the system has reached its new maximum entropy.
A: This is a bit "out there" but it has been conjectured that entropy is in fact what defines our arrow of time. It's not that entropy increases with time, but rather that our perception of time aligns with entropy increase.
Notably, the thermodynamic arrow of time ties the two together by definition: the thermodynamic arrow of time is simply defined as flowing in the direction of entropy increase.
It is arguably impossible to test experimentally whether our perception of the arrow of time is due to the thermodynamic arrow of time, thus potentially relegating this topic to fringe science. It is however well established that information processing and entropy are intrinsically linked: information processing necessitates a certain entropy increase to occur. This is "widely accepted as physical law" by mainstream science; see Landauer's principle.
Taking this one step further, while disputed, it has been suggested that consciousness may essentially be a form of information processing. Connecting this and the Landauer principle, one can conclude that someone like us asking questions about entropy will necessarily live in a world where entropy increases with time (by anthropic principle).
Obviously this topic sits firmly in the gray area between natural science and metaphysics/philosophy, so take this with a big grain of salt, but it is certainly an intriguing explanation for why we see entropy as increasing.
A: This might not be as detailed as you want, but really all the second law says is that the most likely thing will happen. The reason we can associate certainty with something that seems random is because when we are looking at systems with such a large number of particles, states, etc. anything that is not the most likely is essentially so unlikely that we would have to wait for times longer than he age of the universe to observe them to happen by chance. 
Therefore, as you say in your last paragraph, there is no force associated with entropy increase. It's just a statement of how systems will move towards more likely configurations.
For the specific example you give of Joule expansion the (classical) gas molecules are just moving around according to Newton's laws as they collide with each other and the walls of the container. There is no force "telling" the gas to expand to the rest of the container. It's just most likely that we will end up with a uniform gas concentration in the container.
A: From your question, it seems that you call force whatever may be considered as a cause of something happening. However, this it is not the way the concept of force is used in Physics nowadays.
For instance, after Galilei, the uniform motion of a free body far from any other system is an process which does not require a force to happen. At variance, it is the fingerprint of the absence of a net force, according to the Newton's definition of force.
The case of the canister is similar. It is the "closed" configuration which implies the presence of a force to constrain the  gas molecules to remain inside. When you remove the constrain (open the canister) motion of molecules continues without the confining force. The result is their diffusion in the whole available volume just because that is the most probable macroscopic configuration.
