Airplane Mode and Do Not Disturb perform different functions, neither of which are directly linked to push notifications, but can affect them.
Airplane Mode is a quick-fire way to turn off all the wireless radios (cellular, Wi-Fi, Bluetooth, GPS) on the iPhone. You can subsequently turn on the Wi-Fi and Bluetooth radios while still in Airplane mode by going to their respective settings toggles. If you have no data connectivity (i.e. cellular data or Wi-Fi), then you can’t receive any data, whether via push or pull. For more details, see Apple’s support article: iOS: Understanding airplane mode.
Do Not Disturb mode turns off the tones and vibrations from phone calls, SMS/iMessages, and any other push notifications that are set to alert you. However, depending on your settings, some calls may still get through. In the Do Not Disturb settings (Settings → Notifications → Do Not Disturb), you can configure some calls from a specific group of contacts (e.g. your favourites), and repeat calls (a second call within three minutes of the first) to bypass the block.
Do Not Disturb does not prevent your phone from receiving push notifications or pulling data, it merely silences any alarms about the data. It’s also worth noting that alarms set via the built-in Clock app bypass the DND setting.