This got annoying pretty quickly.

When I took a call, or opened up an app after the phone had been resting, it’d ask for my iTunes password. Entering the password or cancelling the dialog would make the request disappear- until next time.

A web search found others with the same problem, so here’s the solution:

One of my apps was updating and had been paused part-way.  The dialog was presumably reappearing every time my wi-fi reconnected. Take a look through your apps and see if any have a static horizontal loading bar across them (it can be difficult to spot), or the word ‘paused’ below. If so, touch them to unpause.

I’d presume that updating the apps via your computer and then synchronising the phone would also work.