Apple - How do I prevent prompts to update to iOS 10

Short answer, No.
Apple really, really, really wants you to upgrade.

Longer answer...
You'd need to be able to block the domains

appldnld.apple.com
mesu.apple.com

on every WiFi access point you meet.

From this article:

iPhone checks whether an update is available through a special XML document at mesu.apple.com. This configuration profile redirects the check to only look for beta updates available for the Apple TV. Since your iPhone is not an Apple TV, the redirected catalog check will make your device “believe” iOS is up-to-date.

The configuration profile is cryptographically signed by Apple (in fact, configuration profile that redirects OTA update catalog through “Internal Settings” will fail to install if it is not), therefore, can be trusted. Other than adding a “Feedback” icon that you can dump into any folder at any time, this configuration profile does not negatively affect your iPhone’s performance or battery life. Don’t worry. It is not possible for your phone to suddenly install tvOS.

Alternatively, you can also block “mesu.apple.com” through your router settings. However, as you connect your devices to Wi-Fi hotspots that you do not have control of, this would be rendered ineffective.

The profile is available here.

You should carefully consider the security implications of downloading and installing a profile from the Internet, but you should also be able to open that profile and inspect its contents before proceeding.