A bit late, but it can be useful for someone else: you must pass the root URL of your app in an environment variable called… ROOT_URL (default is localhost:3000 that can obviously not work, new one is given by Codespace when it launches the app). The simplest way is to launch Meteor with the variable before in command line.
Example: ROOT_URL="https://name-of-your-codespace-an_unique_id-3000.preview.app.github.dev/" meteor
With that I have no more errors in console, and the app refreshes itself at code change (i.e. HMR is working).