I have an application with an export file browser and I want the browser to always open at full screen dimensions. Whenever I resize it and reopen it, it returns to its original (smaller) dimensions. How can I change this?

Might help if you gave some useful information, such as what application, under what window manager, on what version/distro of Linux.

I'm not familiar with the "export" file browser. In konqueror you can save a profile. Select the "Save Window Size with Profile Button".

The you can duplicate the konqueor menu item and change the command from

kfmclient openProfile filemanagement
to
kfmclient openProfile <your_profile_name>

Often you can launch a gui program in the console and enter the --help option to learn options that you can add to the command entry for an gui shortcut or menu item:
example:
did you know that you can convert pdf files to postscript files in the console using acroread!
It has many more options, such as exporting only odd pages, reverse order, etc.

It's an application created in Adobe FlashDevelop and the OS is Ubuntu.