We have a test environment, but we noticed that the tags are firing on test environment even it is not added on the site domain. Under site domain we only added the live domain of the site.
Are you using a load rule to restrict your tag(s)? If so, can you explain how, specifically, it is configured?
As @steve_long mentioned, using Load Rules to control this would be your best bet.
Feel free to read the following support articles on the topic to familiarise yourself:
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