Yes homemade hot sauce just like store bought hot sauce can eventually go bad and actually for some reason homemade hot sauce goes bad much sooner than store bought hot sauce.
If you have homemade hot sauce you should put it in the refrigerator if not using it within a few hours because bacteria can start growing on the homemade hot sauce and cause it to spoil.
If the homemade hot sauce is spoiled it will either start smelling bad or have a terrible taste to it and should be thrown out if it smells bad or tastes bad.
If you keep the homemade hot sauce in the refrigerator in a sealed container it should last from 3 to 6 months before going bad.
You can make the homemade hot sauce last a bit longer by keeping it in a freezer safe sealed container and freezing the hot sauce.
Then just thaw the hot sauce out when you're ready to eat it.
Store bought hot sauce usually keeps for 3 years out at room temperature after opening but for some reason homemade hot sauce doesn't keep as long and needs refrigerated.