I have kept tamari sauce out on the counter or in the pantry without refrigerating the tamari sauce and it's lasted for months.
But most people recommend refrigerating the tamari sauce which helps it retain it's quality and makes it last longer.
I started keeping my tamari sauce in the refrigerator after reading online that it would keep fresher longer and keep it's taste longer as well.
After following that advice I would also recommend putting the tamari sauce once opened into the refrigerator as it keeps it much fresher.