double sum = 0.0;
.
.
.
double average = sum / 12.0

Пробвай да сложиш .0 при делението, а и не виждам да принтиш нещо на конзолата, може да е и от това.

Някак така дали ти е по-разбрано:

import java.util.Scanner;

public class Graduation_2 {
    public static void main(String[] args) {
        Scanner scanner = new Scanner(System.in);

        String name = scanner.nextLine();
        int counter = 1;
        int countExc = 1;
        double sumGrade = 0;


        while (counter <= 12) {
            double grade = Double.parseDouble(scanner.nextLine());

            if (grade >= 4.00) {
                sumGrade += grade;
                counter++;

            } else {
                countExc++;
                if (countExc > 2) {
                    System.out.printf("%s has been excluded at %d grade", name, counter);
                    break;
                }
            }
        }
        if (counter >= 12) {
            double avrGrade = sumGrade / 12;
            System.out.printf("%s graduated. Average grade: %.2f", name, avrGrade);
        }
    }
}