I am not sure if I understood what exactly you are trying to achieve but I will give it a try.
The first problem is that you can't return a value for your outer function within an inner callback function (see https://stackoverflow.com/a/6847754/3173842 for more information on this).
Additionally you are returning the callback function not the result.
This is what I think you actually wanted to do:
var async = require("async");
var listFunctions = {
    test1: function(objectData, callbackData) {
        objectData.name += ":a";
        callbackData(null, objectData);
    },
    test2: function(objectData, callbackData) {
        objectData.name += ":b";
        objectData.foo = "bar";
        callbackData(null, objectData);
    },
    test3: function(objectData, callbackData) {
        objectData.name += ":c";
        callbackData(null, objectData);
    },
    test4: function(objectData, callbackData) {
        objectData.name += ":d";
        callbackData(null, objectData);
    },
    test5: function(objectData, callbackData) {
        objectData.name += ":e";
        callbackData(null, objectData);
    }
};
function abcd(objectData, cb) {
    listFunctions.test1(objectData, function(err, data1) {
        listFunctions.test2(data1, function(err, data2) {
            listFunctions.test3(data2, function(err, data3) {
                listFunctions.test4(data3, function(err, data4) {
                    listFunctions.test5(data4, function(err, data5) {
                        cb(err, data5);
                    });
                });
            });
        });
    });
}
function initalFunction(objectData, cb) {
    var results = [];
    async.each(objectData, function(dataValues, done) {
        abcd(dataValues, function(err, response) {
            results.push(response);
            done();
        });
    }, function(err) {
        cb(err, results);
    });
}
var objectData = [{id:1,name:"abcd"},{id:2,name:"abc2d"},{id:3,name:"3abcd"},{id:4,name:"4abcd"}];
initalFunction(objectData, function(err, response) {
    if (err) {
        console.log(err.stack);
        return;
    }
    console.log("Result:", response);
});
Now lets use async.seq and async.map to simplify this:
var async = require("async");
var listFunctions = {
    test1: function(objectData, callbackData) {
        objectData.name += ":a";
        callbackData(null, objectData);
    },
    test2: function(objectData, callbackData) {
        objectData.name += ":b";
        objectData.foo = "bar";
        callbackData(null, objectData);
    },
    test3: function(objectData, callbackData) {
        objectData.name += ":c";
        callbackData(null, objectData);
    },
    test4: function(objectData, callbackData) {
        objectData.name += ":d";
        callbackData(null, objectData);
    },
    test5: function(objectData, callbackData) {
        objectData.name += ":e";
        callbackData(null, objectData);
    }
};
var abcd=async.seq.apply(null, [
    listFunctions.test1,
    listFunctions.test2,
    listFunctions.test3,
    listFunctions.test4,
    listFunctions.test5
]);
function initialFunction(objectData, cb) {
    async.map(objectData, abcd, cb);
}
var objectData = [{id:1,name:"abcd"},{id:2,name:"abc2d"},{id:3,name:"3abcd"},{id:4,name:"4abcd"}];
initialFunction(objectData, function(err, response) {
    if (err) {
        console.log(err.stack);
        return;
    }
    console.log("Result:", response);
});
If you can change listFunctions from object to array:
var async = require("async");
var listFunctions = [
    function test1(objectData, callbackData) {
        objectData.name += ":a";
        callbackData(null, objectData);
    },
    function test2(objectData, callbackData) {
        objectData.name += ":b";
        objectData.foo = "bar";
        callbackData(null, objectData);
    },
    function test3(objectData, callbackData) {
        objectData.name += ":c";
        callbackData(null, objectData);
    },
    function test4(objectData, callbackData) {
        objectData.name += ":d";
        callbackData(null, objectData);
    },
    function test5(objectData, callbackData) {
        objectData.name += ":e";
        callbackData(null, objectData);
    }
];
function initialFunction(objectData, cb) {
    async.map(objectData, async.seq.apply(null, listFunctions), cb);
}
var objectData = [{id:1,name:"abcd"},{id:2,name:"abc2d"},{id:3,name:"3abcd"},{id:4,name:"4abcd"}];
initialFunction(objectData, function(err, response) {
    if (err) {
        return console.error(err.stack);
    }
    console.log("Result:", response);
});
if not but still need to handle dynamic listFunctions:
function initialFunction(objectData, cb) {
    var list = Object.keys(listFunctions).map(function(name) {
        return listFunctions[name];
    });
    async.map(objectData, async.seq.apply(null, list), cb);
}
I hope this helps.