Will updating to Android 7.0 cause my phone to lose all its application data ? If so, how can I prevent this?
OTA updates don't wipe the device: all apps and data are preserved across the update.

Even so, it's always a good idea to back up your data frequently. As you point out, not all apps support the in-built Google backup mechanism, so it's wise to have a full backup just in case. See our backup tag wiki and top questions to learn about what options are available (whether you've rooted the phone or not).