I would like to know if these medications (for cancer pain) could cause the throat to close up, (or give the sensation of closing up)?

I have not heard of that. If you are having that and it's anything other than completely mild, you should call a doctor or other health professional right away. Don't wait. It could be a serious allergic response, either to the medicine or to something else. Narcotics do depress breathing though, so combined with your throat closing up, that's not very good! Is someone there with you? If not, call someone right away to come keep an eye on you until you can be seen or can talk to a doctor. You may need to go to an ER. Do you have an inhaler or an epipen? Do you have asthma?
Take care!