javascript: how to parse a FileReader object line by line

I guess you should put

document.getElementById('files').innerHTML = 'Paths with more than 240 characters: <br><ul>' + output.join('') + '</ul>';

inside onload callback. Seems that output is not yet populated when you try to use it. So:

reader.onload = function (e) {
    // all your code ...

    // now can safely print output
    document.getElementById('files').innerHTML = 'Paths with more than 240 characters: <br><ul>' + output.join('') + '</ul>';
};

The obvious way (if you can tolerate reading the file all at once) is to split it by newlines.

var lines = text.split(/[\r\n]+/g); // tolerate both Windows and Unix linebreaks
for(var i = 0; i < lines.length; i++) { /* do something with lines[i] */ }
// or in modern JavaScript,
lines.forEach(function(line) { /* ... */ });