Sound is a vibration wave picked up by the auditory circuits of the ear. Vibrations are caused by the expansion and compression cycle of a material, or medium, in air. Events of expansion and compression have varying speeds depending on the material, or medium. The speed creates a frequency of vibrational wave; various speeds create various corresponding frequencies. The frequency (speed of vibration) of the tearing of a dry piece of paper is at a vibrational frequency range high enough to be heard by the human ear.
Tearing wet paper vibrates at a lower vibrational frequency that is not as readily heard by the human ear apparatus. Similarly, vibrational rates that are higher than the range the human ear can register are also not heard. In both cases, however, lower than the range the human ear can register and higher than it can register, vibrations occur and vibrational wave frequencies exist.
While the explanation given in the answer above may have some validity, it does not appear to me to be a complete explanation. To fully answer this question be need to understand better the changes taking place within the paper being torn which generate the noise.
A paper is made of small cellulose fibres tangled together and glued to each other with some chemicals. The tearing of papers results from some fibre being pulled loose from others and some fibres being broken. Both these actions cause the surrounding air to be disturbed with a sudden force causing sound. When a paper is wet the gluing effect of the chemicals holding paper fibres together gets reduced and the fibers get pulled loose more easily creating less noise in the process. Also more fibres get puled out completely rather than being broken. Thus the noise by breaking o fibre is also reduced. As a result the noise mad by tearing of wet paper is less than that of tearing dry paper.
The validity of the above explanation is also supported by the fact that tearing of wet paper requires less force than that for dry paper, and torn edges of wet paper has more uneven edge because of longer fibres being pulled out of the paper, rather than breaking at the tearing line.