Can Botox Cause my Head to Fall Forward After Shots in the Neck?

This would take an incredible amount of botox and It is possible but not likeley that this would occur. .

You seem to be indicating that your muscles in the back of the neck are weak. You're having difficulty lifting your head from a neck flexed position. If the muscles in the forward part of the neck were treated, as they are commonly, then it does not make sense that you are having such diffiuclty.  If you had some Botox in the back of the neck, possibly for a trigger point of Migraines, then yes, you might have difficulty lifting your head up. This will be temporary.

The concern about BOTOX shots causing one's head to fall forward are unlikely depending on where it was given. For cosmetic purposes, BOTOX is uncommonly  administered in minute quantities to bands on the front of the neck.  The muscle being treated is very superficial and has virtually no significant effect on neck movement.  In order to cause the head to fall forward, BOTOX would have to be administered to the muscles at the back of the neck in sufficient quantities to weaken those muscles.  This would not be a cosmetic application.