I mean I doubt he killed Tyrell, but Tyrell did put on the threatening gloves in his apartment, so I'm guessing Elliot wanted to make sure that he walked out of the fsociety building alive. Makes sense he'd go for the gun.

Maybe he used the gun to threaten Tyrell after them finishing their work, and has locked him away somewhere. Maybe he shot to incapacitate him, and Tyrell is passed out in some hospital.