Once a bottle of Tabasco Sauce has been opened it doesn't need refrigeration and can safely be kept in the pantry, on the counter, table etc and it will not spoil.
You can refrigerate the opened bottle of Tabasco Sauce and it will keep the Tabasco Sauce fresher longer but Tabasco Sauces doesn't require refrigeration.
If you don't refrigerate the Tabasco Sauce after opening you may notice that the Tabasco Sauce changes color some but it doesn't mean the Tabasco Sauce is spoiled.
Homemade Hot Sauce needs to be refrigerated but store bought Hot Sauce such as the Tabasco Sauce goes through a process that makes it stay good without refrigeration.