How does adding salt affect the surface tension of water? If it increases surface tension, will it always increase it?

Yes, adding salt to water does increase the surface tension of water, although not by any significant amount.
It is a very common misconception that salt is a surfactant, i.e. a compound that either lowers or breaks surface tension. However, experiments done with salt water show that surface tension actually increases when salt is added to pure water.
As you know, sodium chloride, or ##NaCl##, is a strong electrolyte, which means it completely dissociates into sodium cations, ##Na^(+)##, and chloride anions, ##Cl^(-)##, when placed in water.
It turns out that the strong interactions between the sodium cations and the partial negative oxygen, and the chloride anions and the partial positive hydrogens, although they disrupt part of the that takes place between water molecules, actually strengthen the surface tension of water.
In other words, you get some ionic component to the overall hydrogen bond-dominated interactions from the addition of these cations and anions.
A very common experiment meant to illustrate this concept is done by placing salt water drops on a penny. I’ll post a link to a great article on such an experiment performed by Lucas Cherkewski
http://lucas.ecustom.ca/science-fair/