If I use a bow to play two strings at once on a violin or other string instrument, I don't hear beats occurring. Why's that?
Because the violin is in tune...
In fact, you can often hear a beat frequency between adjacent strings of a violin, and this is used to tune the instrument (once the A string has been tuned to 440 Hz). However, to quote Wikipedia 'a beat is an interference pattern between two sounds of slightly different frequencies.' A violin G string is tuned to 196 Hz, and its neighbour, the D string, is tuned to about 294 Hz. So why would there ever be a beat frequency between the two strings?
Let's examine the spectrum of frequencies from a violin string more closely. Here is a spectrum of the open G string (from here).
There is not simply one frequency peak, but rather a whole range of higher harmonics, at integer multiples of the fundamental G3. (For the reason why, see here.) I'll draw this as a cartoon now, concentrating on the fundamental and the first three harmonics.
Suppose now our violinist starts playing the D string at the same time. The peaks from the D string are shown in orange below (and the musician is playing this string slightly more quietly, to help us distinguish the two.)
The crucial thing to observe is that the second harmonic of the G string, and the first harmonic of the D string, are both precisely at a frequency of 588 Hz. But this is not the case if the violin is out of tune - if it is, a beat frequency is heard between these two harmonics. By carefully adjusting the tension of the G string, this frequency can be reduced to zero - and the violin is now in tune. (Or at least, those two strings are in tune relative to each other...) The effect can just about be heard on violin tuning how-to videos on Youtube, such as this one at 2:17.
You do hear beats if you play the same note on two adjacent strings at the same time. If one of the string is left free and the other is gently touched at the right position to get the other string tone the same as the empty string, as flageolet, the possible beats can actually be used to fine tuning the strings.