Is the iPhone 5s physically similar enough to the iPhone 5 to use a close fitting case?
Per Ars Technica:


  "Every measurement, right down to the 3.95 ounce weight and 0.30-inch thickness, is identical to the iPhone 5."


Visually from looking at photos the ports, buttons, etc. are all identically placed. Also, one of the image captions in that article:


  "The leather cases for the iPhone 5S will also fit the iPhone 5. This should also hopefully mean that third-party cases for the iPhone 5 will continue to fit the 5S."


The only possible issue would be the iPhone 5 LED flash vs the 5s' dual-LED flash. But as long as the cut-out on the third-party case looks like Apple's does (i.e. it goes around the camera and the LED in one cut-out), there shouldn't be any reason they don't fit.