JavaScript is not Java, so System.out.println doesn't have any special meaning. You have two options here: to use console.log(), or to use document.write().
I recommend you use console.log(), as it doesn't mess with the current page's HTML structure:
var favnumber = parseInt(prompt("What is your favorite number?"), 10);
var name = 'JavaScript';
for (var i = 0; i < favnumber; i++) {
    console.log(name + ' is not Java');
}
You'll need to open up your browser's JavaScript console to see those messages.
Using document.write() is a bit more cumbersome:
var favnumber = parseInt(prompt("What is your favorite number?"), 10);
var name = 'JavaScript';
for (var i = 0; i < favnumber; i++) {
    document.write(name + ' is not Java');
    document.write('<br />');
}
Demo: http://jsfiddle.net/HC3Y2/