1. Setting the alert: Go to the iPhone's basic applicaton "Settings" menu and check if "Alert" is set to "On".
After verifying that the "Alert" is "On", go to KakaoTalk's settings, situated in the same menu, to check if "Alert", "Alarm", and "Sound" are "On".
After changing all of the modes that were mentioned above, make sure you put the iPhone on "Airplane mode" for 30 seconds and then restart it.
Then, activate the KakaoTalk application to complete the process.
2. Alert does not appear even after setting: To use the alerts, the TCP port 5223 needs to be open to TCP traffic. If you are connected to Wi-Fi, check with the Wi-Fi administrator if your TCP port 5223 is firewalled.
Refer to http://support.apple.com/kb/HT3576?viewlocale=en_US for further reference.
3. If your iPhone/iPod is "Unlocked" or in a "Jail-break" state: Even though there is no official support for this, try using "pushfix".