Visking tubing seems to be used primarily for removing sugars in aqueous solution. Would it work for removing salinity (on a lab scale) in the same way that semi permeable membranes are used for desalinating salt water on a larger scale?

I don't see why not, but it would be very inefficient. Visking tubing is really just a semi-permeable membrane, so you could certainly use it to separate extremely small ions. You see, it used to separate sugars, but it has many other uses; dialysis centers use visking tubing to purify blood for people who have lost a kidney, but you may see it used in biology classes as a representation for the cell membrane, blood vessels, and digestive system.
The ability of visking tubing to pass molecules and atoms depends on the size of the tubing pores and the size of the molecules attempting to pass through. Similar to trying to fit the large square peg through the small circular hole, large molecules, such as massive starch chains and hulking proteins, will be unable to pass through the tiny holes of semi-permeable tubing. Small molecules like water and glucose can still pass through. When salts are in solution, they break into ions, meaning that molecules like NaCl will break into both Na and Cl atoms. These atoms are more than capable of passing through holes sized for glucose.
The problem you run into is that over time the salinity inside the visking tubing would come to match the salinity outside the tubing. This would have the same effect as just dumping salt water into your water, and pulling out a sample. The salt would just disperse, and the pure water you had before would be salty as well.
In the end, your procedure would resemble fractional dilutions rather than filtering. You would have less salt in the tubing as you ran pure water over it, but your pure water would be just as contaminated.