What are the tradition ways to signify/advertise that a potentially dangerous long-running test is in progress?
In the case of laser safety, there are several kinds of safeguards used. These should all either apply or be adaptable to other kinds of hazards:


Signage --- laser installations require signs indicating the equipment is dangerous, some classification of how dangerous it is ("class 1", "class 2", etc.), and the appropriate protective equipment required when entering the dangerous area (what type of eyewear is needed, for example)
Barriers --- People should be physically be prevented from entering the area except where the signage is present. Doors or gates in the barriers should be locked and only trained and authorized people should have keys to pass through.
Interlocks --- The laser should be disabled if the barriers are opened.
Education --- Anyone with access to the laser area must be educated on the meaning of the signs, the appropriate protective equipment, etc. People who have access to the area around the laser should be educated on why it's dangerous to enter the actual hazardous area and instructed not to enter.


As for a case where an experiment needs to be shut down at the end of the day, I'd suggest to use a computer control to automatically shut it down, and also you as the person in charge, or some other trained person should be required to check that it actually shut down. (Even better, the human check is done first, and the computer only shuts it down some time later as a back up)


  There are two risks I want to avoid:
  1) Harming a bystander who is unaware that the apparatus is performing tests.


This should be manageable with the protections I outlined above.


  2) Accidentally leaving the premises when a long-running test is executing, creating e.g. a fire hazard when the test is left unattended.


This scenario probably requires an interlock to prevent the equipment operating if the attendant leaves. For example a dead-man switch, or some input needed periodically from the user to keep the equipment operating. 

Even better, the equipment should be redesigned to not catch fire.