I don't use Tinder, but I'm trying to answer your questions based on the provided information.
What is the reason for this happening?
Seems you don't have Google Play services installed. Download and install it from Google Play.
Why would Tinder need an active Gmail account on the phone?
As mentioned in the comment by Dan Hulme, "Needing Google Play Services installed isn't the same thing as needing a Gmail account, nor even a Google account." which is partially true regarding Google account (read the next section).
In other words, what Tinder functionality could possibly break?
For Tinder-specific functionality, possibly nothing. However, as already stated from the updated info,
Failed to Enable Push Notifications. Your Google account credentials are invalid.
you won't get any push notification from Tinder (through Google Cloud Messaging). If you don't care about it, then you may just ignore it. However, if you want to have it, then you have to setup a Google account. From Google Cloud Messaging documentation,
"It uses an existing connection for Google services. For pre-3.0 devices, this requires users to set up their Google account on their mobile devices. A Google account is not a requirement on devices running Android 4.0.4 or higher." (Emphasized mine)
Could this explain why I am not getting any matches?
As explained in your self-answer, no, the matchmaking system has nothing to do with Google Play services and Google Cloud Messaging.
Addendum: for devices which don't have access to Google services (Google Accounts, Google Play app and Google Play Services) can try a workaround to download from Amazon App Store (courtesy of Hugo's answer), which seems to be independent of Google services.