If you try to launch AntConc on a Debian-based 64-bit system, you get the following error message (tested with versions 3.2.4u and 3.4.1u):

./antconc3.2.4u: No such file or directory
or 
./AntConc: No such file or directory

The following steps were necessary for me to be able to start AntConc on a TurnKey Linux Server (Debian7, 64-bit) using ssh with X11-forwarding enabled (e.g. PuTTY plus Xming on Windows 8.1).

Important note: Please respect Laurence Anthony’s licensing terms and ask for permission before using AntConc in a server/group environment (see README section ‘LEGAL MATTER’ (p. 11) for details).

1) Activate i386 architecture on 64-bit systems:

apt-get install libc6-i386
dpkg --add-architecture i386

2) Install missing 32-bit libraries:

apt-get install libx11-6:i386 libxss1:i386 libxft2:i386

For Ubuntu-based systems see this post, for other Linux distributions see ongoing discussion on: https://groups.google.com/forum/#!forum/antconc