When an object is animated as per the "Animate Object" spell, or a similar effect, it effectively becomes a creature. This creature is listed in the Monster Manual as an "Animated Object". What happens to the original object when the creature dies? Does the magic cease to function and the object reverts to its non-animated state? Or does killing the monster mean the same as destroying the object?
Objects animated through the 6th-level Clr spell animate objects [trans] (PH 199) use statistics for animated objects (MM 13-14), and, since animated objects possess the type construct, they are "destroyed when reduced to 0 hit points or less" (307).
Since the item is only animated (therefore engaging in battle or whatever with its actual parts) and (except as noted) unchanged by the magic, the destruction of the animated object should, likewise, destroy the object.
For example, if the PCs defeat an animated tapestry by reducing it to 0 hp, the animated tapestry is destroyed. When the animate objects spell's duration expires, damage to the tapestry's isn't, for some reason, healed. It becomes simply a destroyed tapestry instead of a destroyed animated tapestry.