It would appear that I have run into a possible bug in Domain Mapping. Prior to upgrading to PHP5.6, the issue did not appear, but with the upgrade to PHP7, I am seeing a number of mixed content/insecure form warnings in all major browsers.
In short, the issue is related to any attempt to log into a Multisite install via a subdomain of the primary domain or any domain mapped with Multi-Domains. Under those conditions, the browser will throw a warning that the form is attempting to submit via an insecure connection. The page (and the login form) will load via https, but the form will attempt to submit via http.
wp-login.php:1 Mixed Content: The page at 'https://docs.omnifora.com/wp-login.php?redirect_to=https%3A%2F%2Fdocs.omnifora.com%2Fwp-admin%2F&reauth=1' was loaded over HTTPS, but requested an insecure script 'http://omnifora.com/dm-sso-endpoint/1481971418/?dm_action=domainmap-check-login-status&domain=docs.omnifora.com'. This request has been blocked; the content must be served over HTTPS.
and (same error)
Mixed Content: The page at 'https://docs.omnifora.com/wp-login.php?wpe-login=omnifora' was loaded over a secure connection, but contains a form which targets an insecure endpoint 'http://docs.omnifora.com/wp-login.php?wpe-login=omnifora'. This endpoint should be made available over a secure connection.
Troubleshooting So Far
1. Disabled CloudFlare.
2. Forced SSL/TLS through CloudFlare.
3. Forced SSL/TLS through Domain Mapping.
4. Forced SSL/TLS through the backend/server.