How do I disable prompts to sign into the iTunes store?

You cannot remove iTunes Store from the phone, it is a native iOS app. You should not be prompted to sign in to the store unless you are trying to make a purchase. You can go into Settings>General>Restrictions and set up restriction for the iTunes Store. That very well could stop the prompt. At least you can try that. When you set up Restrictions, make sure you write the passcode down, as there is no way to recover/reset that passcode without wiping your iPhone completely.

I have the same intermittent problem. ( I agree I should not be prompted to sign in unless making a purchase, but that's not what's happening.) I haven't been able to pin down a reproducible case. I think the prompt only occurs when unlocking my iPhone (6s, current on IOS 9.3.1). If I dismiss it, subsequent unlocks will eventually stop showing it. Then a day or a week later, I'll see it again.

Sorry, we tried. Someone else may have some ideas.

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Re: how to stop "do you want to sign in to itunes" message