I didn't even know Anton Yelchin was in Green Room until I was checking his filmography yesterday, but I really want to see the movie because of Patrick Stewart anyway.
I don't think they'd recast Chekov given the circumstances, they'll probably start the next movie with Chekov already gone, either passed away during a mission, or reassigned with a different character in his place. I wonder if it's too late to dedicate Star Trek Beyond to him?
For many reasons, I think it would be best to just finish up with Beyond.