By default, it is impossible to load files on callbacks:
The Concept of Callbacks
However, there's a solution that allows you to overcome this problem. Our ASPxWebControl class has the RedirectOnCallback method that allows redirect to another page during a callback.
In our situation, we can create a specific "download" page and call a method which loads a file in the Page_Load event handler of this page.
This sample shows how to load a file in the click of a custom button of the ASPxGridView.
See also:
ASPxTreeList - How to download a file by clicking a custom command button
ASPxGridViewExporter - How to show content document in a new tab