* Megrez is already not shining in our night sky. It's a star in the constellation Ursa Major (the Big Dipper). While it appears to be a single star to the naked eye, it's actually a binary system, meaning two stars orbiting each other.
* Light travel time: Stars are incredibly far away. Even if Megrez were to suddenly stop shining, it would take the light from that star the same amount of time to reach us as it always has.
* Megrez is about 79 light-years away. This means the light we see from Megrez tonight left that star 79 years ago. If it stopped shining today, we wouldn't notice for another 79 years.
In essence, we'd still see Megrez for a very long time, even if it went dark today. We're always seeing stars as they were in the past, not as they are in the present.