I'm having the same problem. I tried replacing the relay with an E-bay purchased unit. And still the light remains on. When I disassembled my original, it appeared physically to be okay.
I haven't had time to debug this further, but I have another piece of evidence that may or may not be relevant. I discovered that with an empty tank of gas, the gauge reads 1/8 full, and AAA take 1 hour to show up with 4 gallons of gas.
So I don't know if the sending unit is hanging up, if it's just inaccurate, or if there is a bad ground connection that may be disrupting the function of the gauge and the LFI light. The gauge will indicate a full tank of gas, while the LFI light is still on.
At least now I have those nice LED bulbs installed, so they aren't creating too much heat when it's on for long periods of time.