mv listener.log listener.log.old
name change doesn't matter, the file is already open. Process still writes to what is now listener.log.old

gzip listener.log.old
(listener.log.old.gz is created and listener.log.old is removed)
you removed the name, not the file. Another name change which doesn't matter.

A file will no link to a directory will be removed upon the last close.

What would probably work:
cp listener.log listener.log.old
cp /dev/null listener.log
gzip listener.log.old