Copy it to your hard drive before installing.

I thought of that, and tried it. But I get errors that indicate that
certain files can't be found. I get a sense that the install program is
looking at my CD drive for certain pieces of the app even though I'm
installing from the hard drive.

You could try changing to root to install. Just open a terminal and type
sudo nautilus. This will ask for your password and everything that you do in
Nautilus (file manager) will have root privileges and permissions should not
be a problem. I don't use Wine so can't check to see if this works.

Never run Wine as root

wiki.winehq.org/FAQ

I have actually not had this problem, but you can get around not being
able to change the content of the CD by copying the cd files to the
desktop or a USB stick, and installing from there.

As mentioned, this method doesn't work because the install seems to look
for files on the CD drive regardless of where I begin the install from.
I haven't tried copying the files to a USB stick, though. I will give
that a try.

There may be hidden files on the disk. Make sure to reveal them
before copying. I have many times installed software by coppying
contents on cd to USB on my wifes netbook. You have to consider there
may be an issue with wine. If you go to wines site, you can check if
others have sucessfully installed your program and what, if any,
workarounds were needed.

I'll check the Wine site and display hidden files, as you
suggest.