How to remove item from array by value?

This can be a global function or a method of a custom object, if you aren’t allowed to add to native prototypes. It removes all of the items from the array that match any of the arguments.

Array.prototype.remove = function() {
    var what, a = arguments, L = a.length, ax;
    while (L && this.length) {
        what = a[--L];
        while ((ax = this.indexOf(what)) !== -1) {
            this.splice(ax, 1);
        }
    }
    return this;
};

var ary = ['three', 'seven', 'eleven'];

ary.remove('seven');

/*  returned value: (Array)
three,eleven
*/

To make it a global-

function removeA(arr) {
    var what, a = arguments, L = a.length, ax;
    while (L > 1 && arr.length) {
        what = a[--L];
        while ((ax= arr.indexOf(what)) !== -1) {
            arr.splice(ax, 1);
        }
    }
    return arr;
}
var ary = ['three', 'seven', 'eleven'];
removeA(ary, 'seven');


/*  returned value: (Array)
three,eleven
*/

And to take care of IE8 and below-

if(!Array.prototype.indexOf) {
    Array.prototype.indexOf = function(what, i) {
        i = i || 0;
        var L = this.length;
        while (i < L) {
            if(this[i] === what) return i;
            ++i;
        }
        return -1;
    };
}