Ah, you are talking about the Wunderground dashboard in WS View.
As far as I know the posting to WU, WOW, WeatherCloud and Ecowitt.net by the console is done with a FQDN (fully qualified domain name) which needs to be resolved by a DNS server.
Again, you are connecting to the internet - this has nothing to do with your console. You read (via WS View) from e.g. wunderground.com what your console has already posted. There may be a cache issue with the app. Maybe clearing the cache of the app will help. That's rather an internet access issue than a weather station (console) issue.
If your IP changes frequently, your provider will still offer the same DNS servers. There is, however, a possibility that the CDN (content delivery network) servers which WU uses are overloaded and the request for WU gets stuck.
It happens quite often to me with WU that my browser displays an error page and only reloading of the web page shows again my station data.
At your console end nothing can be done here.
If your console connected to WU directly via an IP (what it pretty surely doesn't do), it wouldn't be touched by a DNS server issue. And the change of your internet IP for your router has nothing to do with you reaching another web site. The other way around, reaching your IT infrastructure from outside, i.e. from the internet, could be impacted by a frequently changing IP address, even if you used a dynamic DNS service (I guess that's what you are referring to by this no-ip DNS).
Of course, if the change of your IP happens during a request (requesting a web page, e.g. the WU dashboard in WS View), then the display could be stuck as the response wouldn't find the requester anymore. But even with a dynamic IP DNS this would be the case as the dynamic DNS service needs to learn first that the IP has changed. So some disruption cannot be avoided in such scenarios.
