I have a couple of thoughts on this question, and though I admit they are speculations, I hope they are educated ones!
Paper is an interesting material. As to what exactly paper is, Wikipedia tells us that:
Paper is a thin sheet material produced by mechanically and/or chemically processing cellulose fibres derived from wood, rags, grasses or other vegetable sources in water, draining the water through fine mesh leaving the fibre evenly distributed on the surface, followed by pressing and drying.
What holds the fibers together? This is a very interesting question as we still don't have a complete picture of the forces holding these fibers together as this recent (2013) paper paper tells us:
Several mechanisms have been suggested to play a significant role in forming the fibre-fibre bonds that lead to the fibre network that we call paper. In the past, the following possible bonding mechanisms were considered: hydrogen bonds, mechanical interlocking, electrostatic interactions, interdiffusion of cellulose molecules and van der Waals forces. More recently, microcompressions, capillary bridges and stress due to the drying process have been under discussion. What is agreed is that all of these mechanisms contribute in some way to bonding two pulp fibres together. It is still not clear, however, which of these mechanisms dominates or if an interplay of several mechanisms is relevant.
Now, as other folks who have posted answers to this question correctly note, sound is a vibration. When you tear paper, you are breaking the bonding mechanism that holds the fibers together. As you begin to tear, these bonding mechanisms resist and the fibers, on a microscopic scale near where you are trying to tear, get stretched apart from one another as if they were attached to little springs.
Once you actually make a tear, the fibers on either side of the tear rebound backwards because of this spring force and that creates vibrations in the air.
Now when paper is made wet, water, being a polar molecule, finds it easy to insert itself between fibers, since cellulose is also polar. The image below from this link, gives a visual of this:
We must assume there that the insertion of a water molecule, in addition to pushing the fibers further from each other, significantly weakens the "spring constant" between fibers.
I would imagine that this has two effects:
1 - The stretching between fibers is much less when tearing wet paper that dry and this, in turn, reduces the amplitude of the pressure vibrations in the air (sound), and
2 - The water molecule acts as a damping mechanism, or cushion, further reducing the air pressure vibrations that occur during a tear.
The connection between the mechanical vibrations and the air pressure vibrations is, I suspect, also two-fold:
1 -Obviously, the vibrations of the fibers which strike air molecules.
2 - When the fibers are stretched some air molecules probably move slightly into some of the surface interstices between the fibers for a few layers. When the fibers snap back after the tear starts, these air molecules are pushed back out, thereby striking other air molecules.
The strength of the response of the air in both of these mechanisms to the tear will be affected by the strength of the mechanical response of the fibers during tearing.
Finally, I would also point out, though it must be obvious at this point, that the question is somewhat flawed. Wet paper DOES make a sound on tearing. It is just that this sound is much softer than dry paper.