Remove clicked <li> onclick

UPDATE

Plain JS delegation

Add the eventListener to the UL to delegate the click even on dynamically inserted LIs:

document.getElementById("ul").addEventListener("click",function(e) {
  var tgt = e.target;
  if (tgt.tagName.toUpperCase() == "LI") {
    tgt.parentNode.removeChild(tgt); // or tgt.remove();
  }
});

jQuery delegation

$(function() {
  $("#submitButton").on("click",function() {
    var text = $("#item").val();   //getting value of text input element
    var li = $('<li/>').text(text)
    $("#ul").prepend(li); 
  });
  $("#ul").on("click","li",function() {
    $(this).remove();
  });
});   

Original answer

Since you did not mention jQuery

var listItems = document.getElementsByTagName("li"); // or document.querySelectorAll("li"); 
for (var i = 0; i < listItems.length; i++) {
  listItems[i].onclick = function() {this.parentNode.removeChild(this);}
}

you may want to wrap that in

window.onload=function() { // or addEventListener
  // do stuff to the DOM here
}

Re-reading the question I think you also want to add that to the dynamic LIs

li.innerHTML = text;    //inserting text into newly created <li> element
li.onclick = function() {
  this.parentNode.removeChild(this);
   // or this.remove(); if supported
}

Here is the complete code as I expect you meant to code it

Live Demo

window.onload=function() {
  var button = document.getElementById("submitButton");
  button.onclick = addItem;
}   

function addItem() {
  var textInput = document.getElementById("item");  //getting text input
  var text = textInput.value;   //getting value of text input element
  var ul = document.getElementById("ul");  //getting element <ul> to add element to
  var li = document.createElement("li");  //creating li element to add
  li.innerHTML = text;    //inserting text into newly created <li> element
  li.onclick = function() {
    this.parentNode.removeChild(this);
    // or this.remove(); if supported
  }
  if (ul.childElementCount == 0) {  //using if/else statement to add items to top of list
    ul.appendChild(li); // will add if count of ul children is 0 otherwise add before first item
  }
  else {
    ul.insertBefore(li, ul.firstChild);
  }
}

In case you want to use jQuery, the whole thing gets somewhat simpler

Live Demo

$(function() {
    $("#submitButton").on("click",function() {
        var text = $("#item").val();   //getting value of text input element
        var li = $('<li/>')
          .text(text)
          .on("click",function() { $(this).remove()});
        $("#ul").prepend(li); 
    });
});   

If you don't want to write function in javascript, you can use immediately invoked anonymous function like below...

<elem onclick="(function(_this){_this.parentNode.removeChild(_this);})(this);"

I'd suggest simplifying things a little:

Object.prototype.remove = function(){
    this.parentNode.removeChild(this);
};

var lis = document.querySelectorAll('li');

for (var i = 0, len = lis.length; i < len; i++) {
    lis[i].addEventListener('click', remove, false);
}

JS Fiddle demo.

Of course, having done the above, I'd then have to go further (possibly because I like jQuery too much) and also:

Object.prototype.on = function (evt, fn) {
    var self = this.length ? this : [this];
    for (var i = 0, len = self.length; i < len; i++){
        self[i].addEventListener(evt, fn, false);
    }
};
Object.prototype.remove = function(){
    var self = this.length ? this : [this];
    for (var i = 0, len = self.length; i < len; i++){
        self[i].parentNode.removeChild(self[i]);
    }
};

document.querySelectorAll('li').on('click', remove);

JS Fiddle demo.

I know you already received an answer, but back to your original remove function. You have the following:

function remove(e) {
    var li = e.target;
    var listItems = document.querySelectorAll("li"); 
    var ul = document.getElementById("ul");
    li.parentNode.removeChild(li);        
}

Change it to this and you should get what you were trying to achieve:

function remove(e)
{
   var li = e.target;
   var ol = li.parentElement;
   ol.removeChild( li);
   return false;
}