The Living Tribunal does destroy universes. If he deems that a universe has been too "damaged" -- whatever that means -- or is in too much danger from malevolent forces to ever be saved, then he'll destroy that universe himself rather than let it slowly die away.
In fact, you might say that the Phoenix is amongst those guardians of reality -- others including Adam Warlock with the Infinity Watch and Dr. Strange as the Sorcerer Supreme -- who is charged with the duty of preventing just that from happening. The Infinity Watch guards the Infinity Gems and the Power Cosmic, Dr. Strange guards this reality from otherworldly invaders, and the Phoenix coexists with the M'Kraan Crystal to burn away obsolesences within this reality that aren't working. If any of those duties don't get fulfilled, the Living Tribunal might get trigger-happy.