After upgrading my iPhone 6 to iOS 10, iTunes and iPhoto on my computer (mac OS X version 10.7.5) will no longer sync with the phone - any solutions around this? My mac is not capable of moving to newer operating system.
iOS 10 requires iTunes 12.5.1 if you're on a Mac. However, the last version of iTunes to support macOS X Lion is 188.8.131.52, so I'm guessing that's your problem.
If you still need to be able to transfer media such as photos and music between your Mac and your iPhone 6, you can opt to use third party software. One example is iExplorer which happens to still support OS X 10.7 and iTunes 10.
Another possibility, depending on your actual model of Mac, is to install newer versions of macOS via alternative methods. If you want to pursue this, please post another question providing full details of your Mac hardware (including model, RAM, etc).