It’s hard to know for sure, but I would guess not.
We only really see one basilisk in canon – the basilisk in the Chamber of Secrets. Although there are descriptions and allusions to others, we don’t see them in enough detail to make a call on this one. And Fawkes unhelpfully blinds that basilisk and fills the eyes with blood.
JK Rowling’s mythology is not written in isolation; it draws inspiration from a lot of classical mythology. There’s one story about a basilisk – the basilisk of Warsaw – which suggests to me that post-mortem, a basilisk stare isn’t fatal:
He declared that it really was a basilisk; it had the head of a cock, the eyes of a toad, a crest like a crown, a warty and scaly skin "covered all over with the hue of venomous animals," and a curved tail, bent over behind its body.
At this point, the basilisk has been killed and dragged out a cellar. It’s being examined by a physician. He must have looked at the eyes to know they looked like a toad, and he must have survived to tell the tale. Ergo, basilisk stare isn’t fatal if the creature is dead.