Your GFCI will let you know about its displeasure, don't you worry.
Some surge suppressors do indeed transmit some energy to the EGC -- however, this is not a major concern for typical units as MOVs absorb energy in addition to shunting it.
A/V (or other) equipment in metal chassis is not a concern as well -- it is the local equipotential bonding of the chassis that provides protection from EM and RF noise as the EGC is electrically "long" at any significant frequency.
This leaves the issue of devices that either inadvertently or intentionally leak excess current to the EGC -- Tester101 mentioned some "smart" home automation equipment that uses the EGC instead of a neutral return. There are also some pieces of test equipment that leak current to ground as an unavoidable consequence of their normal functioning. The usual sign that an incompatibility with the GFCI protection is present is simply that the GFCI trips (either immediately, or when some poor sod pokes the device) -- either that, or a device that leaks operating current to ground will not function without a way for the current to get back home, of course.
(Of course, Tester's concerns about communicating the situation to the next poor sap who has to work on your house wiring are very important as well.)