Would this pump water up? and if so, how far? I had this idea of an osmotic pump way back in high school and I never got a satisfactory answer if it would work.  If I had this configuration:

Would it continually pump water up given ambient heat so long as the bottom reservoir is full?
EDIT To explain what is happening, there is pure water in each of the dark blue reservoirs, saline in the cyan containers (same concentration of saline in each container) and a semipermeable membrane at the bottom and near the top of each container.
The pure water in each reservoir would be sucked up into each container directly above it due to osmotic pressure (high water concentration flows to lower water concentration), and then dumped out at the top also due to osmotic pressure (saline to air which is almost 0% water concentration). Since the membrane is not permeable to salt, only the water is released from the container into the next higher reservoir.
NOTE that the membrane at the top of each saline container doesn't touch the pure water in the reservoir it empties into.  I'm also thinking that the saline containers may have to be completely filled with extra osmotic pressure to spare to counteract the pure water that sticks to the outside of the upper membrane and cause a reverse osmosis effect.  Other possible tricks relying on surface tension and gravity might also aid in pulling the water away from the membrane.
END EDIT
If enough were stacked, would this allow for transporting water higher than the maximum that trees can transport (about 138m according to this article)?
 A: A quick sanity test for this sort of idea is this: "if it worked, could I use it to construct a perpetual motion machine?" In this case, yes, you could - all you need to do is let the water flow back down from the top reservoir to the bottom again, via a waterwheel, and you'd have an endless source of work without putting any energy into the system. This means that your idea breaks the first or second law of thermodynamics somewhere along the way, and the only remaining task is to figure out where exactly this happens.
The problem in this case is that you're relying on the air being "almost 0% water concentration" (i.e. this machine is only supposed to work on dry days, when it's not very humid.) You're expecting that when the water moves out of the upper membranes it will stay in a liquid phase and drip down into the reservoirs below. 
However, when the water molecules move from region of liquid water into a region of dry air, they don't stay liquid but become vapour. We call this evaporation. Whether water can evaporate through the membrane depends on the balance between the vapour pressure of water in the air and the osmotic pressure of water in the solution. If the air is dry then water molecules will indeed move out (very slowly) from the upper membranes, but they will turn into water vapour. They won't condense back into the upper reservoirs, because the air is dry and doesn't want to give up that moisture. In fact, the upper reservoir will be losing water to evaporation itself, at a much faster rate.
So when the air is dry enough for water to pass through the upper membranes, all that will happen is that the water in all of the reservoirs will evaporate. You will not see an accumulation of water in the upper reservoir.
A: It won't work.  It is true that a semi-permeable membrane can raise a column of salt water until the pressure (due to the column) matches the osmotic pressure across the semi-permeable membrane.
The problem comes at the top of each level: how do you envision that water gets out of the salt column into the next higher pool?
If the fresh water in the upper pool actually touches the membrane at the top of the column, it will flow into the column, increasing the water level, increasing the pressure at the bottom, forcing water down into the bottom pool.  Thus you end up draining the upper pool into the lower one.
If it doesn't actually touch the membrane, you seem to be assuming that for some reason water will drip out.  But it won't.  If you have a film of water on the other side of the membrane, the osmotic pressure will suck it into the saline.  (In fact, water will be sucked into the saline from the air, as the vapor pressure above the membrane will be lower than the vapor pressure above water, creating net flow of water into the saline.)
