First things first, make sure your Flutter project is all set up with Firebase, and that Firebase Authentication is ready to go.

• Follow the Firebase documentation to add Firebase to your Flutter project.
• Enable the sign-in methods you need (like Email/Password, Google sign-in, etc.) on the Firebase console.

Next, let's add the necessary dependencies in the pubspec.yaml file:

dependencies:
  firebase_core: latest_version
  firebase_auth: latest_version

Then, run flutter pub get to install the packages. Easy peasy, right?

Now, let's initialize Firebase in the main.dart file:

import 'package:firebase_core/firebase_core.dart';
import 'package:flutter/material.dart';

void main() async {
  WidgetsFlutterBinding.ensureInitialized();
  await Firebase.initializeApp();
  runApp(MyApp());
}

Time to create functions for signing in, signing up, and handling user authentication:

import 'package:firebase_auth/firebase_auth.dart';

final FirebaseAuth _auth = FirebaseAuth.instance;

Future<User?> registerWithEmailPassword(String email, String password) async {
  UserCredential userCredential = await _auth.createUserWithEmailAndPassword(
    email: email,
    password: password,
  );
  return userCredential.user;
}

Future<User?> signInWithEmailPassword(String email, String password) async {
  UserCredential userCredential = await _auth.signInWithEmailAndPassword(
    email: email,
    password: password,
  );
  return userCredential.user;
}

Let's set up an authentication state listener to keep an eye on user sign-ins and sign-outs:

_auth.authStateChanges().listen((User? user) {
  if (user == null) {
    print('User is currently signed out!');
  } else {
    print('User is signed in!');
  }
});

Firebase usually refreshes the user's ID token when it expires. But, just in case, let's add some code to handle scenarios where this might fail:

Future<void> signOut() async {
  await _auth.signOut();
}

Future<void> refreshToken(User user) async {
  try {
    await user.getIdToken(true); // Forces token refresh.
  } catch (e) {
    await signOut(); // Sign out if token refresh fails.
    print('Token refresh failed: $e');
  }
}

void checkAndRefreshToken() async {
  User? user = _auth.currentUser;
  if (user != null) {
    user.getIdTokenResult(true).then((idTokenResult) {
      DateTime expirationTime = idTokenResult.expirationTime as DateTime;
      Duration diff = expirationTime.difference(DateTime.now());
      
      if (diff.inMinutes < 10) { // If token is about to expire in next 10 minutes
        refreshToken(user);
      }
    });
  }
}

Finally, let's use Timer to regularly check and refresh the token:

import 'dart:async';

void main() {
  Timer.periodic(Duration(minutes: 5), (timer) {
    checkAndRefreshToken();
  });

  runApp(MyApp());
}

This way, your tokens will be refreshed periodically. And there you have it!