For anyone interested, I experienced this issue (Bravia KDL40EX503U, iPlayer previously worked OK, then stopped at "Loading iPlayer" logo stage, Demand5 still working on the same TV).
Factory reset (Settings - General Settings - Factory Settings) fixed it.
I'm having issues with my KDL-32EX723.
Getting black screen with small 'progress' spinner in top right corner. Only way to get the TV unstuck is to unplug it. This has become such a regular occurrence in the last month or so that I've now rearranged the plugs so this reset procedure is easier!
From what I've observed it is iPlayer related. Happening either when content has just finished or when first requesting the service from the 'Internet Video' screen. Therefore, from what I can see, it's probably a bug in TV firmware triggered by something (recently new?) in iPlayer menu 'scripts'.
With the exception of the buffering issues last year (been and gone) iPlayer has been fine on this TV up until a month or two ago.
Sort it out Sony. Not only are your functionally poor and technologically backward products being outstripped by the likes of Samsung and LG but long time Sony loyal customers like me are being poisoned, by you, against your brand by unacceptable bugs. I have an AV technical and computer programming background so have an informed context to look at this from. It's not good enough. Given what I'm experiencing from your product my next TV purchase will not be a Sony. You're losing me.