I think my issue may be similar to this post but I'll explain my situation. I have an older home and the living room is on a circuit that contains 2 non-grounded outlets and a wall switch.
The wall switch operates one of the outlets, the other is always hot. When the switch is OFF, the light plugged into that outlet turns off, but my contactless voltage tester still beeps on both the white and black plastic insulated wires attached to the terminals on that outlet.
The other outlet (the always hot one) appears to have fabric-insulated hot and neutral wires attached to the top terminals (right and left side respectively), and another neutral wire (this one plastic insulated) connected to the bottom left terminal. In the box behind this outlet, a fabric-insulated wire is connected to a black plastic-insulated wire with a wire nut. I'd like to replace the "always hot" outlet with a GFCI outlet but the GFCI outlet doesn't like that second neutral wire being attached to the "load" terminal. When I removed it from the load terminal, it worked and the little light was green, but as soon as I flipped the wall switch on, the GFCI outlet tripped.
If any of that makes sense, I'd love some assistance figuring out what is going on.