JSFiddle - React, Tailwind, and code Playground

JavaScript

var objTeste = [{
    "id": "03",
    "nome": "teste03",
    "pai": {
        "id": "02",
        "nome": "teste02",
        "pai": {
            "id": "01",
            "nome": "teste01"
        }
    }
}]

function recursiva(obj) {
    var _obj = {};
    for (var k in obj) {
        if (typeof obj[k] == "object" && obj.hasOwnProperty(k)) _obj[k] = recursiva(obj[k]);
        else if (k != 'id') _obj[k] = obj[k];
    }
    return _obj;
}
var novo = objTeste.map(recursiva);
console.log(JSON.stringify(novo));