When the wick runs dry you won't see visible flame but the deposits in the wick will be glowing like charcoal. When it finally goes out nothing is left but harmless white ash.
Refill, wait 15 minutes for the wick to prime and you're good to go.
We make regular use of a kerosene heater in our home. A real pain to replace the wick every couple of years - looking up the right cross reference and finding one. You’d think by now I’d have a few extra wicks and igniters on hand, but no. I wonder if this guy poured kerosene directly into the flame? We never fill our heater unless it’s been turned off for at least 15 minutes.