Is it easier or harder to move moist air with a fan? From what I understand, the molecular weight of water is lower than the molecular weight of nitrogen and oxygen. So, shouldn't it be easier for a fan to move moist air than dry air of the same temperature?
If so, why does it feel as if your fans aren't blowing air on hot, humid days?
I originally thought that it was because the water on your skin was not evaporating into the already saturated air, but someone offered a rebuttal that was more or less 'That's not how that works' 
Will someone clarify, please?
 A: "Thats not how that works" ;)
There's actually two questions, so I will try to answer them one after the other:
1.) The molecular weight of nitrogen and oxygen isn't particularly relevant for gases in the pressure/temperature ranges you are talking about. A guy named Amedeo Avogadro found out, that each gas we know of exacts the same pressure when it is heated to the same temperature inside the same volume.
What this means is, that not the molecular weight is relevant, but rather the amount of molecules/atoms inside a specific volume at a specific temperature. 
This is the ideal gas equation p * V = n * R * T, with n = amount of atoms/molecules and R = a constant for unit conversion reasons.
Imagine a gas molecule hitting a solid surface - the force it applies to the surface is J = m * V ( https://en.wikipedia.org/wiki/Impulse_(physics) )
If the molecule is heavier, its speed V automatically is slower, so that J always has the same value.
Now, what does this has to do with moisture?
If you put 1m³ of dry air into a box, at a temperature of 20°C and a pressure of 1 bar and put it onto a weighing scale, you will get the value of ~ 1.293 kg. If you add a bit of water vapour to that air (up to the point, where the air is saturated), the amount of molecules inside the box increases and so does the weight: we either have to release the pressure out of the box or the pressure inside the box will rise.
Lets assume that the pressure stays the same (since you are talking about a fan in your home, and your doors/windows aren't sealed airtight): put your box onto a weighing scale again and what do you get? The weight actually has decreased, because we let more molecules out of our pressure release than we put in with our water!
If the Blades of a fan (= solid surface) hit the air molecules, the fan motor needs to overcome this pressure. Now with the vapor we have less molecules in the air than before, therefore the fan needs less force to move them.
This effect is barely noticeable in day to day applications like a fan at your home. It does play a significant role in industrial applications though.
2.) So, why does it feel bad, on humid days?
The surface of your skin always has small drops of sweat on it, even on cold days. You can only saturate the air around you until a certain point of humidity is reached. When the air already is humid, less sweat can vapourise. ( https://en.wikipedia.org/wiki/Evaporative_cooler#Physical_principles )
This feels bad, because the evaporative cooling effect is the main means of temperature regulation of a human being. So, even if your fan is blowing more m³ of air to you, it can only evaporate a fraction of your sweat compared to dry air.
In fact, this is one of the main reasons air conditioners feel so good - because they not only cool the air down, but also dry it. This feels nice on the skin, but is bad for all "lubed" parts of you body, eyes, mucous membranes, etc. because they need the water to work properly.
