Hitting a string of a violin or a guitar will cause that string to vibrate, but after short time the amplitude of the vibration will decay, consequently the produced sound will die out.
I suppose this decay happens because of friction with air. If this is true then how much longer will the string keep vibrating in a vacuumed room? Any way to estimate this? Are there other effects that cause the damping?
The instrument is designed to make sound. The loss of energy is not due to friction, but to emitting sound. In a vacuum, a suspended plucked guitar would ring for minutes, not seconds.
The loss of energy to sound is a direct mode-coupling, and it takes energy away regardless of internal friction. But you can estimate the degree to which internal friction is important by comparing the ringing time of materials with negligible internal friction for sound propagation--- crystalline materials like metals--- vs. complex polymers like wood or plastic, where the propagation of sound leads to losses because the restoring forces are partially entropic.
For a steel body guitar with steel strings, there is no plausible avenue for sound modes to decay fast in the resonator, because the steel is a crystal material. To get an estimate for internal losses in wood, compare the resonating time for a note/chord steel body guitar and a wood guitar in air. The wood guitar frictional losses are estimated by the decay time of the tone in a wood guitar vs. steel body.
Here is a steel body demonstration: http://www.youtube.com/watch?v=tVx62GpWKOE
I don't hear noticably less decay in the metal, so I assume that internal losses in wood are small compared to radiated sound energy.
Friction doesn't just come from air, it comes from two sources.
For the vibration to work in the first place, the string must be stretchable. As it is stretched, the tension increases. When it vibrates in standing waves it oscillates between a high tension, no velocity state to a low tension, high velocity state.
Although the system looks different, we can treat this fairly similarly to a normal dampened harmonic oscillator system. You can say the string starts stretched to some length, $l_o$ and tension $F_o$. It is not uncommon to treat this system with a drag force proportional to "velocity", although I would suggest a superficial definition of velocity in this case, which is the rate of contraction or elongation of the string over time.
$$F = F_0 - k (l-l_o) - c \frac{dl}{dt}$$
This equation, however, is not a complete differential equation. This is because I'm using it in an analogous form to a mass on a spring, and what to use in place of the mass is non-obvious. I won't go into that because I'm not sure how much detail is wanted.
Basically, the energy is still dissipated as heat in the string. The heat is stored there unless it radiates out. The string will eventually stop oscillating, although it will last longer than if it were in air. Obviously, no music is produced unless you consider the vibrations in the structures of the violin and whatever else it's touching music
Well, as far as my intuition goes air friction should not cause much of damping in the vibrating string of violin, though it surely plays a small role in damping. For speeds of the oscillator not so large (do not cause turbulence) certainly a friction force exist in viscous medium, like air, which is proportional to velocity as $ f = -bv $ where $b$ is constant of proportionality which depends on medium and object.
Major damping as I believe should be due to heating of the string. Other source of damping will be transfer of energy to instrument (as Georg pointed).
