It causes way more traffic for the DNS server to use a shorter TTL, so yes, it does incur more DNS traffic. In Germany some providers will disconnect you regularly if you stay connected for too long.

Some ISPs require changes ever 24 hours and will disconnect you if needed. Also, if you set DNS to cache such a short amount of time that you can react to that in 5 minutes, you will incur way more DNS traffic which can become a problem when your site is busier. Also, even if your DNS TTL is set to a super short value, a web search suggests to me in practice there will likely be downstream clients and networks that ignore it and won’t really update in such a short time frame.

Even in an ideal DNS setup, you’re probably going to have downtimes whenever your dynamic IP changes. If only because some ISPs even force-disconnect you after a while to change your address.