Is it safe to open a file that is being written by a running script?

Reading the file is safe, although double clicking you mentioned will probably open the file in some editor that will offer you an option to make changes and save them. Missclicks happen, so I recommend commands that won't even let you change the file.

These are examples that work in terminal; they will only read the file:

cat file
less file
less +F file
tail -n 5 file
tail -f file

As long as you are not writing to it, it should be okay.

However, I would recommend using

tail -f log_file

in another terminal.

This command will "follow" the file log_file and write the newly added content as soon as it is updated by the script.