Can this Code be Improved to Avoid Memory Leaks or be More Stable?

Question

There are a couple things you can do but I see no reason for memory leaks.

I have commented my edits.

var keys = [];

// Integers takes less space than strings, and we might as well put it in an array already
var shortcut = [
    17, // ctrl
    16, // shift
    83  // s
];

window.addGlobalHotkey = function(callback, keyValues){
    if(typeof keyValues === "number")
        keyValues = [keyValues];

    // Unfortunately, because you need the cb argument we can't define it outside the addGlobalHotkey method.
    var fnc = function(cb, val){
        return function(e){
            keys[e.keyCode] = true;
            executeHotkeyTest(cb, val);
        };        
    }(callback, keyValues);

    window.addEventListener('keydown', fnc);
    return fnc;
};

window.executeHotkeyTest = function(callback, keyValues) {
    var i = keyValues.length;

    while(i--) {
        // If key is not pressed, we might as well abort
        if(!keys[keyValues[i]]) return;
    }

    callback();
};

window.addEventListener('keyup',function(e) {
    // Remove key instead of setting it to false
    keys.splice(e.keyCode, 1);
});

addGlobalHotkey(function() {
    var x = getIn("What Task?");

    // If x is empty there's no point in resuming
    if(!x) return;

    // If-statements are faster than switches, just saying.
    switch(x)
    {
        case "a":
            //...
        break;
        case "t":
            //...
            break;
        default:
            //...;
    }
}, shortcut);

function getIn(text){
    var x = prompt(text);

    // Altered if statement
    if(typeof x === 'string' && x.length > 0) return x;
}

The most potential leak is the fact that the array keys will only grow as you press more different keys. Hence the .splice() method.

Edit

I have reviewed your code again. I couldn't help noticing that you create a new onkeydown listener whenever you register a new hotkey. I've changed that. Try the following code instead, and see if it still runs slow.

/* VARIABLES */

var keysPressed = [];
var hotkeys = [];

/* EVENT LISTENERS */

// Listen for keydown once instead for every hotkey
window.addEventListener('keydown', function(e) {
    // A button was pressed
    keysPressed[e.keyCode] = true;
    checkHotkeys();
}, false);

window.addEventListener('keyup', function(e) {
    // Remove key
    keysPressed.splice(e.keyCode, 1);
}, false);

/* FUNCTIONS */

function checkHotkeys() {
    var i = hotkeys.length;

    while(i--) {
        executeHotkeyTest(
            hotkeys[i].callback,
            hotkeys[i].keys
        );
    }
};

function addGlobalHotkey(callback, keys) {
    if(typeof keys === "number")
        keys = [keys];

    hotkeys.push({
        callback: callback,
        keys: keys
    });
};

function executeHotkeyTest(callback, keys) {
    var i = keys.length;

    while(i--) {
        // If key is not pressed, we might as well abort
        if(!keysPressed[keys[i]]) return;
    }

    callback();
};

function getIn(text){
    var x = prompt(text);

    // Altered if statement
    if(typeof x === 'string' && x.length > 0) return x;
}

/* OTHER */

// Integers takes less space than strings
var shortcut = [
    17, // ctrl
    16, // shift
    83  // s
];

addGlobalHotkey(function() {
    var x = getIn("What Task?");

    // If-statements are faster than switches, just saying.
    switch(x)
    {
        case "a":
            //...
        break;
        case "t":
            //...
            break;
        default:
            //...;
    }
}, shortcut);

Edit #2

I've found the issue!

So we've been using the keycode as index for the array - a numeric index. These are not associative. Have a look at the following example.

0: 'abc',
1: 'def',
2: 'ghi'

Delete 1

0: 'abc',
1: 'ghi'

Delete 2

0: 'abc',
1: 'ghi'

By converting them to a string, your array becomes an associate array and the right keys will be deleted.

// Listen for keydown once instead for every hotkey
window.addEventListener('keydown', function(e) {
    // A button was pressed
    keysPressed[e.keyCode.toString()] = true;
    checkHotkeys();
}, false);

window.addEventListener('keyup', function(e) {
    // Remove key
    delete keysPressed[e.keyCode.toString()];
}, false);

Check out the working demo: http://jsfiddle.net/rQePB/1/