Nerve damage can be caused by a dental injection if the dentist hits a nerve during the dental injection.
However it's rare that a dentist will cause any nerve damage when giving you a dental injection because they are highly trained and qualified and are very careful when inserting the needle to give you the dental injection.
I have heard of a few cases where nerve damage resulted due to a dental injection but most dental injections are done without a problem.
I've had a few dental injections myself and never had any nerve damage but it's certainly possible.
Make sure you go to a good dentist and don't always go for the cheapest dentist.
As long as the dentist is qualified and well trained to do the dental injections then you should have no problems with nerve damage.