Hi qwerty_zxcv,
welcome to the forum
Updating a div several times won't cause any memory leak. Each browser has their own way of handling memory cleanup.
I'm not aware of any memory problems at all in Firefox.
However, there is a "well known" memory problem in IE caused by circular references. A circular reference happens when
1. Your javascript code has a reference to a DOM element(example a tag)
2. The DOM element got a reference back to your javascript code.
This typically happens in event handlers. When this happens in IE, the memory won't be released until the browser is completely closed. And you if you do a page refresh, the script will use more and more of your available memory.
You will find a lot more on this topic by making a google search on the words
ie memory leak javascript
But generally, updating a div several times shouldn't cause any memory problems. If you experience these kinds of problems, it's not caused by that, but by some other problems in the code.