Artists and Illustrators frequently use social media, particularly Facebook to show off new art. This is done for a few reasons:
1.) To advertise their illustration to potential clients (they may be friends with many Art Directors and this is a tried-and-true way of keeping them in touch with new work)
2.) To show off new works to collectors who may purchase the work they see.
3.) To present the world with lovely things.
When you see cool art, DO:
1.) Like it: likes are free still and easy, and posts with lots of activity seem to get shown to more people since it is viewed as valuable content and Facebook likes showing content that is engaging. Just liking a post can help the artist out in spreading the art around.
2.) Share it: ok, this takes more work and if you're an artist also a bit of swallowing your pride, but sharing work obviously multiplies an artist's reach.
3.) Compliment or discuss or share a story in the comments, ask questions (but note the Don't list below).
Note that two of the three purposes artists might share work are business-related. Because of that, there are a number of things that you can innocently do that can mess up the artist's ability to accomplish these tasks. Since the comments section can be a free-flowing peanut gallery, it might not occur to you that you are not helping the artist, but unfortunately, you aren't.
When you see cool art, DON'T:
1.) Criticize the artist publicly in-thread. First, your criticism might be valid, or not. Either way other viewers will read your criticism and it could diminish their enjoyment of the art--and if your criticism is off-base, unfairly so. If you'd like to critique the work, DM the artist. If you don't want to do that (the direct way of reaching the artist) then why exactly did you want to criticize in public?
2.) Make fun of or joke about the art. Most artists don't take themselves that seriously, but when you free-associate something you think is funny about the art and point it out in the thread, you are making it easy for everyone else to make the same association. If this association detracts from the art or lessens its impact, you've hurt the artist a little.
3.) Post your own art in the thread: perhaps the artist painted a cool Star Wars piece they are trying to show off to attract interest from people in the industry, and you did a cool art of the same character, too. Don't post your version in the comments--you are hijacking their thread and views. Or, their art reminds you of something someone else did (maybe even better), don't post that art in the comments. Again, DM is your friend if you want to have that conversation with the artist.
Again, most of the reasons artists are sharing art are business-related: trying to gain work or trying to make sales. Since that is how an artist eats, it's important to them, even if social media reach offers only a small effect in this regard. So when you lessen the impact of the art, or innocently ridicule or overshadow it, you inadvertently hurt the artist's efforts at self-promotion, which most artists find hard enough to do as it is!
Most of these tips also can be ported over to Twitter and Instagram, but at least the comment threads aren't as visible in those formats. Nevertheless, good etiquette helps artists. You want to help artists, don't you?