What is the significance of cigarettes in ashtray at opening and closing credits of “The Big Sleep?”

The simple answer would be "because it's noir and it looks good". (And I think it's the right answer).

The more complex would be something like waves and smoke in the beginning and end in the Death in Venice (1971) so it’s a sign of mortality and fragile of human life that can be easily put out like a cigarette.

Two cigarettes are a sign of the human relations where two individuals can burn next o each other started with the same match (passion) which end in demise of them both.