When your stringing up your fly rod & getting your gear together before you head down to the stream, take a quick moment to inspect the line guides as you pull your line through them. Guides can get worn, if the rod is old enough & that can cause spots on the guide which are not smooth. This will cause friction that will not only hamper your casting, but will also slowly damage your fly line.
Also be sure to check for bent or damaged guides as well. Rods get leaned against cars, trees, brick walls, boat seats, etc & all that can lead to a guide that's bent out of shape or damaged. Again this will interfere with you casting & add undo friction to your fly line. A line guide can also develop a burr if mistreated. As you can imagine, that would be murder to your line. If you discover that you have any damaged line guides, they're going to have to be replaced.
Be sure to examine your reel as well. You want to make sure that that the area where your line exits the reel is in good shape with no burrs or dirt. This can happen easily & will hurt your line. Think about it: you set your rod down to untangle the line or take a picture of the fish you just landed. Now, what part of your rig is the making contact with the ground - the reel. Fly reels come in contact with dirt, mud, water, rocks, etc as we shift them around & prop them against trees & other things throughout the course of a day's fishing.
One other thing anglers do that damages fly lines, is standing on it. I'm guilty of this myself. It happens innocently enough: your standing in a boat or on shore & stripping line out & back to you. Soon, a pile of fly line begins to form in loose coils at your feet. You take a step to the left or right, maybe just shift your feet, & you end up standing on your fly line. You're so focused on the fishing that you never even notice. Meanwhile, your crushing your line down into the mud & dirt, or worse something hard like a tree root or the hull of your boat. Those muddy boots of yours are doing things to your line it was never designed to handle.
Paying attention to some small things can truly extend the life of your fly line.