Currently there is an IP setup to a domain.

I am creating a new site for this customer on a different server and different domain.

they want http://domain.com/promotion

to go to the IP of the server that i'm using. How can DNS be setup so that the URL works properly?

You can't point folders to different IP's. You could redirect to another URL though.
It's better to create promotion.domain.com if you want it on a separate server.

That will be impossible. If you want to do that you will need to reconfigure your web server little bit. I suppose working with third level domain name would be much better idea for you.

Or you make a subdomain and you point to this one.

You can create a subdomain and redirect it by editing the DNS zone file.

or, You can use a simple URL redirection using .htaccess file. If you are using cPanel then go to the control panel>> Redirection and redirect http://domain.com/promotion to ip address.

Or a redirection php script