It looks as though you've seen Michael C's response to "Why is my Nikon D750 not autofocusing?". If you've not already double checked all of those settings though, it's worth making sure that none of them could be the reason that autofocus isn't working, even if you don't think there's any possibility that you changed them.
The reason Nikon give that warning is because setting the camera mode to "M" retracts the screw that drives the autofocus to the lens. If the screw is left engaged, then turning the focus ring of an AF lens by hand will also turn the screw which can damage the motor. It seems that AF-S lenses have a different internal design which allows them to bypass the screw when the lens is set to "M", so it doesn't matter so much if the screw is retracted or not.
You can certainly enquire about having your camera repaired, but if as you suspect the damage was caused by ignoring (or missing the significance) of a warning that's printed in the guide supplied with the camera, it's unlikely that Nikon would agree to fix it under warranty (based on my reading of the UK (my region) repair centre T&Cs). You should be able to ask for an estimate first if you'd like to get an idea of the cost before proceeding, though.
