I'm personally not always sure what the protocol is. I've been to places where there is a rack you can stack things on and others where someone may possibly tidy them up that day.
May just look to see what others are doing if I'm not sure.

As for Admiral's Club lounges, until recently, I wasn't aware that the silver plastic cutlery wasn't washed and reused. I always left it on a plate assuming it was of greater value than the average white plastic ones.
Do try and throw napkins and those polystyrene cups in a bin though.