I have Fedora 6 running Apache/PHP/MySQL and Python 2.5.
I can make a web page that submits data to a Python app and it works, see below.
Web form.
<form name="form1" id="form1" method="post" action="/cgi-bin/software_proc_codes.py">

Python code.
#!/usr/bin/python
import cgi
print "Content-Type: text/plain"
print "Hello, world"

The problem I am having is how to call a compiled Python code.

I have tried changing the web form to call
/cgi-bin/software_proc_codes.pyc and failed.

I have removed the "#!/usr/bin/python" from the script, compiled and tried the following.
#!/usr/bin/python
import /var/www/cgi-bin/hello.pyc

Do I need to set something up in Apache to handle .pyc?

Check out the mod_cgi module in httpd.conf.
I haven't used Apache in a while, and I can't recall how you configure module behaviours. I use Lighttpd at home, and I everything is just handled in the lighttpd.conf file.