You do not need to refrigerate Tabasco Sriacha or even regular Tabasco Sauce.
Both Tabasco Sriacha and regular Tabasco Sauce does not need or require refrigeration and can be left at room temperature for a year and sometimes longer.
Tabasco Sriacha is a simple hot sauce with ingredients which include red jalapeño, garlic, vinegar, sugar, and salt, along with thickening agents and preservatives.
Tabasco Sauce that has not been opened will last for 5 to 10 years and possibly longer.
Once opened the Tabasco Sauce will usually keep for about a year but it's best to use the Tabasco Sauce within 6 months because it does lose it's freshness.
You can refrigerate opened Tabasco Sauce to keep it fresher longer but it's not required to refrigerate Tabasco Sauce unless you want too.
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.