I wanted to disable the Gmail app because I don't use it and something on my phone keeps hogging all my bandwidth. When I disabled the Gmail app it warned that "some apps may not function correctly" if it is disabled. I haven't noticed anything. Which apps might not function correctly when the Gmail app is disabled?
It is a standard message to warn that dependent applications will find some functionality missing. Few apps are dependent on gmail. Most other email apps would work fine in its place. The only deficiency that I noted was that the gmail app can accept HTML content sent (shared) from other apps. Other email app turn the HTML content to plain text.
The "some apps may not function correctly" warning is a standard thing that pops up when you disable stock apps. You shouldn't experience any issues for the most part, the only instance where I could possibly see an issue is when using google's other email client "inbox". But even then it's its own standalone app and they use google play services to connect with google and it's services. So as long as you're not using any apps that for some reason directly link to gmail (there shouldn't be any as long as you're not using some weird xposed module or something) in some way you'll be fine.