Yes Tabasco Sauce can be left out at room temperature even after opening the bottle of Tabasco Sauce.
I've never refrigerated any Tabasco Sauce I have had and it's never went bad.
I've kept a bottle of Tabasco Sauce that I opened out on the counter for 6 months before and it was still good to eat.
Tabasco Sauce does not need refrigeration like most other homemade hot sauce.
Homemade Hot Sauce should always be kept refrigerated but Tabasco Sauce does not need refrigeration unless you want to refrigerate the Tabasco Sauce.
The opened bottle of Tabasco Sauce should last at least a year before it goes bad although it could still be good longer.
Most people will eat or use the Tabasco Sauce sooner than a year anyway.