I did confirm that "Home Sharing ... is checked". I have played a song on my iPhone while it is plugged into power and has just synced and it doesn't update the play count. Also, as noted above the iPhone does update the play count in iTunes when the phone pairs with bluetooth in my car. However it only updates a single song-the song playing at the time I leave the house, and presumably my wifi network. Also, my iPad doesn't update play counts except if I plug it into a power source it will update one single play count, the first song it plays. After that, if I play a 2nd song or 100 more songs, it doesn't update the play count past the first song.
"Share my library on local network" is turned off in iTunes preferences as I read that might make a difference. However, I have the same problem with or without that box checked.