When does cooked unrefrigerated salmon go bad?

For food safety reasons, potentially hazardous foods, such as cooked salmon, should not be left unrefrigerated for more than an hour - maybe two. So much depends upon the ambient temperature and the type of microbes that are present.

The length of time that it would take to get to the "yechhh" stage will be longer - but is again dependent upon ambient temperature and the microflora that are present. Salmon that is properly cooked, which means rare, will begin to get gamy overnight. Smoked salmon lasts somewhat longer.

The only salmon that could actually take days of non-refrigeration would be some that has been properly processed/smoked/dried to reduce the pH and/or the water activity or if it has been specially processed and hermetically sealed.

A Russian proverb tells us that fish and visiting relatives both start to stink after 3 days. Without refrigeration, a lot less.