Simply count the number of attempts and exit the loop once it reaches the threshold. Like that:
int attemptsNum = 0;
final int maxAttempts = 3;
do {
System.out.print("Guess a number between 1 and 10: ");
user = input.nextInt();
if (user > comp)
System.out.println("My number is less than " + user + ".");
else if (user < comp)
System.out.println("My number is greater than " + user + ".");
else
System.out.println("Correct! " + comp + " was my number! ");
} while (user != comp && ++attemptsNum <maxAttempts );
if (attemptsNum == maxAttempts)
System.out.println("You loose. The number was :" + comp);
solved Limit number guessing game to three attempts