...is it better to open the output file once, in the beginning and keep it open ..., or open and close it whenever I have something to print into it?

Opening is a system call. Closing results in system calls. Both of these acts consume some amount of time greater than zero. The significance of the amount of time each call takes is something that one could only quantify by knowing how many times the open/close cycle are happening in your script, and how much time your script is spent inside of these calls.

This important information that reveals the significance of the open/close calls within your runtime can be determined by using Devel::NYTProf. But if your script is taking 15 minutes to run with your data set, before using NYTProf, create a sample data set that's about 10-20% of that size while remaining representative of your real data. Then profile, and see where your problems are.