How to temporary block any internet communications of iPhone? Is it possible to press some button and temporary turn iPhone into normal phone, i.e. make it transmit nothing over data channel?

I have such button on Android -- to disable "mobile data". Can I do the same on iOS?
You can turn off all data transfer over the mobile network by disabling Settings->Cellular->Cellular Data. It still does WiFi then, which you can disable with Settings->Wi-Fi.

Alternatively, you can disable all communications by activating Airplane mode.