Here is the crux of the problem. Because most late model vehicles are "prewired" for such things as Navi units and On-Star, the power amplifiers are powered on anytime the ignition is on, whether or not the radio is actually turned on. In fact, most GM units are powered even when the key is off!
In most cases (not all), when the radio is off, the low level preamp circuitry contains numerous transistor junctions. These rectify the induced Rf, and what you end up with is obvious.
Sometime, applying beads to the interconnect wiring and the amplifier will fix the problem. However, in some non-premium sound system cases, the radio and the power amp are on the same chassis, and it is impossible to filter the interconnection.
You can also get rectification in the final amp transistors, and if enough signal is present, you'll hear that too.
To tell which is which, pull the fuses for the radio. Remember, there may be more than one fuse (radio and its associated amp). If you still have the problem, it is rectification in the finals. If the problem goes away, the fix should be obvious.
You didn't say what your vehicle was (is), but if it is a GM, there is a tech bulletin about this very issue. Although it covers On-Star equipped vehicles, the fix is the same.