The gravitational pull of a large number of asteroids perturbs a pulsar's motion to a degree that is detectable through precision timing of millisecond pulsars. The result is a low-frequency, correlated noise process, similar in form to the red timing noise known to affect canonical pulsars, or to the signal expected from a stochastic gravitational-wave background. Motivated by the observed fact that many millisecond pulsars are in binary systems, we describe the ways in which the presence of a binary companion to the pulsar would affect the signal produced by an asteroid belt. The primary effect of the companion is to destabilize the shortest-period orbits, cutting off the high-frequency component of the signal from the asteroid belt. We also discuss the implications of asteroid belts for gravitational-wave search efforts. Compared to the signal from a stochastic gravitational-wave background, asteroid belt noise has a similar frequency and amplitude, and is similarly independent of radio frequency, but is not correlated between different pulsars, which should allow the two kinds of signal to be distinguished.