After googling for an answer and not finding one, I took it upon myself to figure out how to get AOEII:FE running in playonlinux, after enjoying many hours of AOEII:Conquerors on my laptop. After installing AOEII and AOEII:C, I only needed to modify 1 file.
Download the Forgotten Empires manual install package from here:http://www.forgottenempires.net/install/ (bottom of page)
Extract the contents into the installation folder for AOEII (mine was '~/PlayOnLinux's virtual drives/AgeOfEmpire2/drive_c/Program Files (x86)/Microsoft Games/Age of Empires II'
Close PoL then copy '~/.PlayOnLinux/shortcuts/Age Of Empires II : The Conquerors' to '~/.PlayOnLinux/shortcuts/Age\\ Of\\ Empires\\ II\\ \\:\\ Forgotten\\ Empires'
Open ~/.PlayOnLinux/shortcuts/Age\\ Of\\ Empires\\ II\\ \\:\\ Forgotten\\ Empires in the text editor of your choice and change the last 2 lines from
cd "/home/chris/.PlayOnLinux//wineprefix/AgeOfEmpire2/drive_c/./Program Files (x86)/Microsoft Games/Age of Empires II"
POL_Wine "age2_x1.exe" -nostartup $@
cd "/home/chris/.PlayOnLinux//wineprefix/AgeOfEmpire2/drive_c/./Program Files (x86)/Microsoft Games/Age of Empires II/age2_x1"
POL_Wine age2_x2.exe -nostartup "$@"
**Make sure to change the path to the actual installation folder or it won't work.
When you run PoL, you should have Age\\ Of\\ Empires\\ II\\ \\:\\ Forgotten\\ Empires on the list. At this point, you should be able to run the game.
If PoL crashes instantly after running the game, you may need to delete (or rename) '.PlayOnLinux/wineprefix/AgeOfEmpire2/drive_c/windows/system32/dpwsockx.dll'
If you want to enable fullscreen, click configure, hit 'configure wine' under the wine tab, uncheck 'emulate virtual desktop' on the graphics tab.
This was all I needed to do to get it working, but YMMV.