It depends on if it's a dry fly or wet fly (floating vs. sinking). If it's a wet fly, and assuming no current like in a pond, you can start to move it.

In a pond setting, I usually just let a dry fly sit unless it's an imitation of something that should move on the water like a mouse or frog. I'd let a frog site for a little bit though.

Excerpt from Robert Crais' "The Monkey's Raincoat:"
"She took another microscopic bite of her sandwich, then pushed it away. Maybe she absorbed nutrients from her surroundings."