I have recently claimed 2 mobile phones from insurance which is a false claim. I mean that the phone was not lost but still I made a claim, saying that I have lost a phone. So I want to ask whether should I pay back the amount of the false claim or would I ignore this, because I still paid my insurance fees and claim fees.

Please help me, because I don't want to get caught or punished in Akhirat/ grave.

As you said it was false claim therefore this act was haraam. Yes you must pay back the amount of the false claim. Yes you are right you are paying insurance fees but as we know that you are paying for right claim not for false one.
We are representative of Islam we must understand our responsibility