Try to leave your finger on the sensor a bit longer than the usual, for about 30 seconds. You won’t have to do this every time, just once. After that, your Touch ID will work normally.
For some users, the problem was in the iPhone case. Try to remove it, and see if it will change something.
Click on Settings, and tap on Touch ID & Passcode. What you need to do is check if App Store or iPhone Unlock is on.
If there is a new version of iOS available, download and install it. Once you upgrade your device, Touch ID will probably work just fine.
Before you start with resetting the device to factory settings, you should back up your files and data. Navigate to Settings > General > Reset > Erase all Content and Settings. Once you choose that action, you will have to confirm it once the message appears on your screen. When the task is done, follow the steps to set your iPhone 6 Plus as a new device.