Single instrument, fixed location
Musical instruments absolutely can and do cause destructive interference. It happens all the time. Usually, it's only noticeable if the two notes are in tune within a few Hertz, or even a few decihertz.
This causes the audience to hear the interference as "beats", where the magnitude of the sound goes up and down at a noticeable frequency. Piano tuners use this to tune pianos: this task is made easier since the speed of the beats is equal to the difference (so two strings at 100 Hz and 100.5 Hz will "beat" every two seconds), the strings for a particular note are largely identical, and they have extreme control over the pitch of the string being tuned. A slightly out of tune piano will have noticeable beats.
The more complex the tone, the harder it is to hear the beats. It is most noticeable in high-pitched instruments emitting purer waves (e.g., flutes). It can be a real problem in recorder consorts, to the point where being precisely in tune is more painful than being a few cents (hundreds of a half-step) out of tune. In electronic music, where musicians have much more control over the pitch and shape of the wave, some artists have utilized this as a specific effect.
The interference itself can also become a third voice; in classical harmony, this reinforces the chord. For example, the fifth of the scale is 50% above the fundamental; for a chord centered on 100Hz with a fifth at 150Hz, there is a 50Hz beat tone--which is the octave below the fundamental, and therefore sounds "in tune"! This is one extra reason that anharmonic chords sound "more crunchy"; the interference tone itself is dissonant as well.
If two instruments (or more--but it's most obvious with two) are very nearly in tune, the audience will hear beats as well. Typically most obvious in the upper register with a pure tone, which means it's mostly heard with flutes and piccolos, soprano singers, and the like.
(This leads to the musician's 'joke': "How do you get two sopranos to sing in tune?" "Shoot one.")
Sometimes it is best to intentionally play out of tune, 10 cents or more, so that the beats are not as obvious.
Reverberation and acoustics
In a well-prepared concert hall, the space itself is constructed in a way to provide multiple bounces for the sound to arrive at every audience member. This is on purpose, to reduce the occurrence of destructive interference. Some halls have well-known "dead spots" where performers cannot be heard, or where audience members cannot hear; modern acoustic theory uses diffusers to avoid this.
Contrariwise, in an unprepared space (e.g., a flat rectangular closet with hard walls), it is easy to hear the interference caused by reflection, and you can set up standing waves where you can increase or decrease the amplitude by moving your head around.