I have to wholeheartedly agree with Keven. Fruit juices should be strained before shaking; apart from anything else, the second strainer will slightly raise the temperature of a drink you have so purposely just chilled. If you are getting ice shards in the glass, then you need a higher quality Hawthorne strainer. I never get ice shards. I would have to buy a cheaper Hawthorne strainer to be able to do that. The only good reason for fine straining is after muddling.
The temperature slightly increasing from the second strainer is almost impossible to notice and seems like a minor reason to not fine strain cocktails that call for it.