Does 'throw new Exception' require exit()?

No, code after the throw statement is not executed. Much like return.

No, code after throwing an exception is not executed.

In this code example i marked the lines which would be executed (code flow) with numbers:

try {
    throw new Exception("caught for demonstration");                    // 1
    // code below an exception inside a try block is never executed
    echo "you won't read this." . PHP_EOL;
} catch (Exception $e) {
    // you may want to react on the Exception here
    echo "exception caught: " . $e->getMessage() . PHP_EOL;             // 2
}    
// execution flow continues here, because Exception above has been caught
echo "yay, lets continue!" . PHP_EOL;                                   // 3
throw new Exception("uncaught for demonstration");                      // 4, end

// execution flow never reaches this point because of the Exception thrown above
// results in "Fatal Error: uncaught Exception ..."
echo "you won't see me, too" . PHP_EOL;

See PHP manual on exceptions:

When an exception is thrown, code following the statement will not be executed, and PHP will attempt to find the first matching catch block. If an exception is not caught, a PHP Fatal Error will be issued with an "Uncaught Exception ..." message, unless a handler has been defined with set_exception_handler().