Martin Splitt from Google replied to a question on LinkedIn about what we should be concerned about when it comes to rendered and indexed. One of the questions was about network requests. Martin replied, "maybe number of network requests a bit, but also not too much."
Martin doesn't think you need to work much about cost of rendering, CPU usage, and server stuff but he said maybe, just maybe, network requests, in his response. He then added, even for network requests, he wouldn't worry about it too much.
Here are those posts on LinkedIn:
Network requests can be made to GET information from a server. But not only GET, also POST, PUT, PATCH, and DELETE.
There is a 429 server status error for too many network requests as well.
Forum discussion at LinkedIn.