It is not a fuse, it is a relay.
- TNS controls parking lights, tail lights and running lights.
- Not sure what TNS stands for but my guess would be "Taillight, Night Running lights and Stop light"
- Normally the TNS relay is powered by the taillight fuse and is energized when the headlight switch is turned on
Having said that I'm not sure why it would be cycling when the ignition switch is off. Could be a short to ground somewhere in the system. If you have had any work done I would have it rechecked, especially if you swapped out radios or added extra lighting somewhere.
Other things you can try:
- Pull the taillight fuse. The TNS should not have power and the lights will not work.
- If the TNS is a removable type of relay pull it out and see if you still hear something clicking
- With the relay pulled check the base. The coil side should have 12vdc to ground on one side. the other side should not have continuity to ground with the headlight switch off.
- If there are other relays in the box of the same type, try swapping them out and see fir the problem follows the relay.
While these things will not fix the problem it may help diagnose the underlying problem.