Are there any security issues inherent in setting up a mapping between a domain name and dynamic IP (i.e. what dynamic DNS services like DynDNS or No-IP do)? Or are the only security risks related to what comes after (opening up ports, running an ssh server, etc.)?
I don't believe there are security issues, but if you're planning to use this for web hosting, you have to be aware that most ISPs disallow this (you'll need to check with your ISP to be sure).
Also, a dynamic IP is not good for an email server (in case you're planning to run an email server with that IP) as you'll get blacklisted in a very short time.