Are there any plans from Microsoft or Azure to stop or limit support for services running on .NET Core 2.1?
You might keep an eye on https://azure.microsoft.com/en-us/updates/. Not likely that your existing applications will be impacted until an announcement is made to the public.
What limitations or issues might we face if we continue using .NET Core 2.1 on Azure?
The limitations of using end-of-life frameworks are well documented in previous announcements, like this that "we'll no longer provide patches or customer service for .NET Core xxx". Similar statements can be found in .NET 5/6/7 end-of-life announcements as well.