Seriously. The NSA has become a FUD (fear, uncertainty, and doubt) breeding ground in which computer knowledge flies out the window.
Important things to know:
1. NOTHING can be downloaded on your iPhone without physical access to the device or the credentials to your Apple ID.
2. Assuming you don't "jailbreak" the phone, ONLY apps from Apple's App Store can be installed. This means that Apple approves every single app you put on your device, and you can delete these apps instantly.
Note that if you have a phone owned by an institution, things are slightly different. Your IT department might be able to download apps onto your device remotely, but the likelihood that they'd be able to download anything malicious is VERY small. They'd still only be able to install Apple-approved apps, since jailbreaking is illegal.
So, long story short, all of this crap about NSA remote surveillance is just that - crap.
But if you want to wear a tinfoil hat, please take a picture so I can see it 🙂