You can try and do nothing about it:
public static void exceptionTest() {
    makeNullPointer(); //The compiler allows me not to check this
    try {
        throwAnException(); //I'm forced to handle the exception, but I don't want to
    } catch (Exception e) { /* do nothing */ }
}
Bear in mind, in real life this is extemely ill-advised. That can hide an error and keep you searching for dogs a whole week while the problem was really a cat(ch).  (Come on, put at least a System.err.println() there - Logging is the best practice here, as suggested by @BaileyS.)
Unchecked exceptions in Java extend the RuntimeException class. Throwing them will not demand a catch from their clients:
// notice there's no "throws RuntimeException" at the signature of this method
public static void someMethodThatThrowsRuntimeException() /* no need for throws here */ {
    throw new RuntimeException();
}
Classes that extend RuntimeException won't require a throws declaration as well.
And a word from Oracle about it:
Here's the bottom line guideline: If a client can reasonably be expected to recover from an exception, make it a checked exception. If a client cannot do anything to recover from the exception, make it an unchecked exception.