When you add a text widget to a sidebar in Wordpress the widget includes this code
How do I remove this without editing the core?
FYI - I think this line is located here: wp-includes/default-widgets.php (line 383)
You can use output buffering (http://php.net/manual/es/book.outcontrol.php) to get the whole generated page in a variable and remove the offending before echoing.
But it seems overly complicated for this case. You can just copy all the widget code and create a new widget (in a plugin or your functions.php) without that hardcoded div. It's really not that much code.
Also, any reason to remove that div that can not be solved in css?
As per the comment, in the case it is probably better to just remove or replace the class name using jQuery then a php function, since that class is hardcoded anyhow.
To remove it.
To set the css to hide it.
To change the class name
If you're looking to remove the actual div with the textwidget class applied to it but not it's content, this is a simple method using jQuery: