Would it be possible to discover an asteroid belt around another star? With existing or planned technology (e.g. the ELT or the JWST), would it be possible to infer the presence of a ring of asteroids similar to our own asteroid belt? Or would any signal be drowned out in the noise?
It's currently very hard to impossible to observe asteroids around other stars.
Asteroids are bodies with a size of at least several metres to a few hundred kilometres in diameter. There's three main ways to detect bodies:

by their characteristic thermal emission
by obstructing the light of their central star
by reflecting light of their central start to us

Either of those three methods is extremely challenging for bodies of that size: they are not very numerous and their thermal emissions is in the mid-IR (depending on proximity of their host star) and is easily drowned in the emissions of their host star.
Obstruction of the direct light of their host star (transit method). This relies on their geometric cross-section which is very small. We can detect reasonably transits of fraction of 0.001 or possibly with quiet stars and good conditions down to even 0.0001. An asteroid will cause a smaller signal.
They are so small that any reflected light is also very hard to detect even with interferometric imaging methods. This has only been proven possible for some objects at quite a separation to their host star and which are much larger. The reflected light is also much smaller at those separations, and given their size so that we cannot make direct observations.
So in summary: we are currently at the edge of detecting planets of Earth size and are approaching Mars and Mercury with the transit method - both are considerably larger than asteroids.
Thermal emissions are mostly used for dust. Single asteroids have a much smaller total surface and thus are not big enough to cause a significant IR excess at their typical emission wavelength (even to indicate the presence of an asteroid belt with more than single object).
Detection of asteroids via astrometry is unlikely even in the near and mid-term future. We currently only have technology to detect (rather distant and larger) gas-giants around nearby planets. The best we have for astrometric detection currently is the Gaia satellite which has detected a 10 Jupiter-mass gas giant in a 22 year orbit as a companion to Beta Pictoris.