When I startup Lightroom CC (2015.8 release) on OS X 10.11.6 (and many previous versions) I get a warning that Signed Resource Files Have Been Modified as the program attempts to connect to the internet. Any thoughts on what is causing this and how it can be fixed?
That error message is coming from your operating system. It means that it detected that the application has been changed from its official (i.e. signed) version somehow and it prevents it from connecting to the internet. It is a measure against malware. This can be malicious (you might have malware on your system) or disk corruption. Be very wary of many "system optimization" utilities that you see all over the internet promising to make your mac faster. They are all terrible and at best don't do that much damage. I would try reinstalling Lightroom.
I have uninstalled and reinstalled Lightroom and am still receiving the same messages. These messages are being reported by Little Snitch on some (but not all) of Lightroom's connections to the internet.