Would tetanus injection cause fever?

Yes. For Adacel (the adult tetanus and whooping cough booster), fever incidence is 1-5%. Interestingly, there is some evidence that taking tylenol (acetaminophen) before a vaccine in hopes of preventing a fever can actually decrease the immune response to the vaccine - which isn't a good thing. So the fever could signify you're having a good immune response... But that's not conclusive.
It Can. Most vaccines can cause a minor fever and the tetanus vacccine is not different. Not a problem.