Will closing off air vents that are not desired cause more air to come through the ones that are open?

Yes you can do this. But depending on the size of your vents, you can decrease the airflow through the AC unit and may cause freezing of the coils and/or overheating. If I were you, I wouldnt duct tape the vents closed though. I would spend a few bucks and get installable vents with shutters on them to open/close the individual vents. If you ever wanted to take the duct tape off, it would leave glue on everything and would just look terrible.