If the fabric is mounted as a wall or ceiling covering, as specified in International Building Code 2003, section 803.4, then the fabric only is required to have been tested and given a Class A rating according to ASTM E 84. This will be true for typical permanently-installed fabric -- stapled over furring strings, or attached via a track fabric system.
If the fabric is mounted otherwise, as covering an acoustical panel, then NFPA 701 testing applies instead the the ASTM testing. That's where the "701" comes from in "Guilford FR701®," which is used a great deal for commercial acoustical panels. The entire assembly must be tested under NFPA 701, unless it covers less than 10% of the room wall and ceiling area (which would qualify it as a "decorative finish"). Technically, acoustical panel manufacturers must provide proof of this testing.