If I was running a DOS program in windows xp and i needed to reinstall
the operating system, could i simply copy the file contents to the new
installation of the OS, or are there registry entries that are
implemented into the OS?  Do DOS programs even utilize the registry?

There was no registry in DOS.Sometimes the program would require
entries in autoexec.com

DOS wouldn't use any registry entries unless you had set the program
to run a specific way within WinXP (ie: full screen, Compatibility
mode etc..) in which case it may have to store some of these settings
in the registry.

The previous comment mentions Autoexec.com, however the file was
called Autoexec.bat.  I would be very surprised if anything in this
file was required though, as it was used as a list of commands to run
when a PC booted to DOS.

Try copying the folder content and the shortcut. It also might use
other directories, but it's uncommon.

If you don't have another PC, you could try it with DOSBox, which
emulates 286 PCs. Or commercial apps like VMWare or Virtual PC.

DOS programs don't use the registry. Most Win16 programs neither.
Win32 console applications can use the registry, for example "Command
prompt" is not a DOS application.

To back-up the DOS application you only have to copy the folder, and
check the shortcut. It is a .pif file. You need to copy it, and/or
write down the values. Check the directories you have to back-up. It
might use other directories for data, etc.

If in the "Program" tab, [Advanced] button has something different
than "%SystemRoot%\SYSTEM32\AUTOEXEC.NT" and "....\CONFIG.NT" you also
have to back them up. DOS programs might need specific entries in
both.

I believe that the three previous comments are overly complicated. DOS
programs can be copied to and fro between operating systems with no
problem - to try it, simply copy the Folder with the DOS program onto
a floppy (or CD), then pop it on another computer and see if it works.
Bets are that it will.

DOS does NOT utilize the Windows registry, nor require any registered
DLLs (which usually prevent Windows applications being copied to and
from different computers / OS installs). You should be fine.