The dynamic product variables are based on css classes, which tell the built in javascript where to update the price.
https://exp-resso.com/docs/store/tags/product.html#dynamic-product-variables
To get it working, simply wrap your price in a <span>
element, like so:
<span class="store_product_price">{price}</span>
Also note there was a bug in 1.6.0 which caused this to break in some situations (though you mentioned you are running 1.6.1 so this shouldn't be an issue).