New (sealed) chili sauce doesn't need to be put in the refrigerator. After unsealing, do you need to put it in the refrigerator after the lid is tightened?
It is safe to keep your chili sauce in the pantry when the package says it. With the lower water content and due to the sauce being highly processed it's not necessary to store it in the refrigerator. Most sauces are made from vinegar with high acidity and higher salt content that prevents them from attracting bacteria etc.
But most companies recommend storing your chili or hot sauce in the refrigerator. This prevents the sauce from color changes and helps to preserve the long-term quality of the sauce.
There is no health risk, leaving the sauce in the pantry but for an optimal sauce with fresh color and taste, I would store it in the refrigerator. Also, it lasts a lot longer when stored refrigerated.
For homemade chili sauce, I found this article interesting if you would like to read more about it. But there is also already a similar thread about the storage of hot sauces under Should I refrigerate hot sauce?