Open a console (in KDE, Konsole would suffice).

If you can, goto where the file's located - eg. cd ~/downloads/

Then type in
ls -l LimeWireLinux.bin

That should give you something like this (results may vary):

-rw------- 1 myname mygroup 6303174 Jul 7 22:29 LimeWireLinux.bin

If the first part that you see is exactly the same as what's shown (ie. -rw--------)
type in:

chmod +x LimeWireLinux.bin

exit, and then try running it.