This actually looks like an iOS bug. There are quite some related issues that I cannot put all of them in the problem subject, so I am going to describe the background now.
Update by the end of 2020: Based on the follow-ups and answers, it seems that this problem still exists in iOS 14 and also occurs on Apple Watch.
iOS 10 has a region-specific function, that allows Chinese iPhone users to turn off Internet access completely, not just cellular data access. This function is associated with the iPhone model, not with cellular network operators, actual location of users, or language settings. This means it applies to iPhone bought in a Chinese Apple Store, a Chinese authorized store, or from an authorized Chinese website. Not sure if it has to be China, though. Anyway, a US model will not support this function, which means you can only turn off cellular access if your iPhone is a US model. Here is a third-party description of this function but it is in Chinese: 具透 | 你可能不知道，iOS 10 有一个中国「特供」的联网权限功能
Screenshots for this function:
The disturbing problem that I actually wanted to ask, happens this way:
I have two iPhones, an iPhone 5 (Chinese model, for China Mobile) and an iPhone 5s (US model, for T-mobile, and also works with AT&T), both running iOS 10.2.1 (latest version when I was encountered with the issue). None of them has a SIM card inside, which means I cannot use cellular data on either of them.
I backed up the iPhone 5 and recovered the backup on the iPhone 5s. The motivation is simply to migrate from one to the other.
For most of the applications and settings it worked well.
However, there are two applications, which when I open them, say "Wi-Fi is turned off for name of the application". (I cannot provide a screenshot for this because this dialogue only appeared once for impacted apps, even if I re-install the apps.) The dialogue pointed me to Settings, but the US iPhone 5s does not have Wi-Fi on/off switches in the Settings, and cellular data was set to "on". Now, these application cannot use the Internet -- they just say "No Internet connection".
What I tried:
Turn off cellular access of impacted apps. Problem persisted.
Turn on cellular access of impacted apps, again. Problem persisted.
Delete the app from the home screen, and reinstall from the App Store. Problem persisted.
Go to Settings/General/Reset, and reset the network settings. Wi-Fi passwords were removed, however, when I reconnect to Wi-Fi, those apps still cannot use Internet.
Checking if anyone has had similar issue, and if possible, how to fix this without resetting everything on the iPhone 5s.
I am not sure why some apps are impacted while some others are not. Impacted apps are:
CamScanner. It uses Internet to sync scanned files.
不背单词. It is an English learning app that uses Internet to download vocabulary.
This might be relevant and might not be. It could be some settings on the iPhone 5 making such difference but I no longer have access to those settings on the iPhone 5 now.