Does anyone that go/goes to my gym know/knows if they're open?
What to use and why.
Also is this correct: did anyone that go to my gym knew if they're open?
Does anyone that go/goes to my gym know/knows if they're open?
What to use and why.
Also is this correct: did anyone that go to my gym knew if they're open?
The main clause is "Does anyone know if they're open?". The subordinate clause is "that goes to my gym".
Take the main clause first. When you have an auxiliary verb, the main verb (or non-auxiliary verb) is always non-finite (which means no inflection for person, tense or number). So, we say "she can be" (not "can is"), "she can know" (not "can knows"), "she did go" (not "did goes"), "does she know" (not "does she knows"), and so on. It follows that "know" is non-finite in this sentence, and therefore "knows" would be wrong. ("Knows" is a finite form used for the 3rd person singular present tense.) "Know" here is a bare infinitive. "She does know", "anyone does know", "does anyone know", etc.
Now take the subordinate clause. The antecedent of the relative pronoun "that" is the word "anyone". "Anyone" is grammatically singular (as also reflected in your correct choice of "Does" to start the sentence). So the finite verb "goes" must agree with the 3rd person singular - "go" would be incorrect. Unlike "know", "goes" isn't governed by an auxiliary. "Goes" is finite and must agree with its subject.
Finally, your sentence *"did anyone that go to my gym knew if they're open?" is ungrammatical because the main clause is "Did anyone knew" (which corresponds to a declarative or non-inverted clause *"Anyone did knew"). Again, the presence of the auxiliary verb "did" means that the verb it's governing has to be a bare infinitive "know". So, we always say "she did know", "she didn't know", "did she know", "did anyone know", etc - never *"did knew". In the subordinate clause, the verb should be "goes" (as before). If the state of being open or closed is in the past, "they're" becomes "they were".