What I see is: $ ./LiveSuit.sh
Starting x86/LiveSuit.
x86/LiveSuit: error while loading shared libraries: libaudio.so.2: cannot open shared object file: No such file or directory
..which is because tinycore does not have a nas extension.
If I make my machine look like an i486, then I get: $ ./LiveSuit.sh
Error: unknown architecture i486
..so this would indicate the minimum hardware for LiveSuit (not tinycore) is i686, exactly what machine are you trying to use?
Edit: As a hack, if you copy libaudio* from the debian package libaudio2 into the filesystem and load the qt-4.x-base extension, LiveSuit will run on tinycore as long as you have at least an i686.