The short answer is no. You probably are asking because you've heard people say "long time no see" and by analogy you wanted to use a different verb. "Long time no see" is a set phrase so you can't actually change the wording. That said, native speakers CAN say "long time no eat" as a joke because it's obviously wrong, but both of them know it's wrong and know the meaning, but as a non-native speaker if you said it, people might think you actually believe it is correct grammatically speaking. It's a strange situation where the only time you can say this is if other people know that you know that it's grammatically incorrect.

I'm afraid not. "Long time no see" is an idiom. It's a stock phrase. It is used exactly as given, those four specific words. It is not a pattern you can use with other words in place of "see."

People will still understand what you mean, but no, it definitely isn’t correct. The correct form is “I haven’t eaten it for a long time.” I hope it helps :)