Try to keep your environment and yourself properly hydrated.
(It's probably easier to keep your room/apartment/house whatever hydrated; drinking enough fluids while playing games like the ones you mentioned can cause other issues ...)
If you're not blinking much as it is, dry air and dehydration can make things worse.
Think about getting a humidifier that you can run during long sessions, and do what you can to keep fluids in without constantly having to go to the bathroom.
If you wear glasses or contacts and have the choice, wear glasses.
Having contacts in will just make the symptoms you describe worse, because now you have to keep both your eyes and the contacts moist.
gaming.stackexchange.com/users/8366/Dave DuPlantis
As silly as it sounds I will sometimes blink one eye at a time for a good long blink
and I find I feel a lot better.
gaming.stackexchange.com/users/2351/Tyler Szabo
You could give a try to F.lux .
This small software allows your computer to automatically decrease the gamma value of your screen, to match the luminosity of your environment.
And guess what?
It's free of charge!
Good point:
it's now avalaible for Windows, Linux and Mac!
gaming.stackexchange.com/users/7098/Lysarion
I doubt that you stop blinking entirely, but it is common to blink less when using a computer.
You have a few options: Use glasses instead of contacts.
Contacts tend to dry out eyes.
Another poster mentioned this.
Use wetting eye drops (NOT VISINE).
Some common brands are Celluvisc and Blink.
The thick and gooey drops will keep your eyes wet for longer sessions, but leave residue around your eyes if you don't wash it off.
Humidifier would help if you are in a really dry area.
Though you probably don't want to add too much moisture in an area with a lot of electronics.
It's possible that if your eyes are drying out that fast that you have some bacteria around your eyes that either slow the flow or reduce the quality of your tears.
Daily warm compress and increasing fish oil intake can help with this.
You might want to see an optometrist if you suspect this.
Of note, my wife is an optometrist.
I have asked her a lot of questions about dry eye, since I have it, but I'm too lousy a patient to regularly follow her instructions.
Let me know if you have any other questions and I'll see about asking her.
gaming.stackexchange.com/users/11962/Perishable Dave
I've never personally used them myself, but have you heard about GUNNAR Optiks glasses?
They claim to help with eye fatigue, and other problems caused by starting at computer screens.
It's not exactly a cheap option, they run around $80 and up for a pair.
gaming.stackexchange.com/users/4239/Doozer Blake
Get some eye drops.
I found some in China town that are menthol (sounds weird but they are awesome) and they work perfect.
gaming.stackexchange.com/users/1396/Abe Miessler
From Dave DuPlantis's answer keep yourself hydrated make sure to drink plenty of fluids whilst gaming (I keep a couple of pint glasses of water on the table beside).
Then, keep a few slices of freshly cut onion beside you.
That's guaranteed to keep your eyes moist and force you to blink regularly.
Remember to cut a new slice of onion a few times during your session.
gaming.stackexchange.com/users/263/GAThrawn
Sometimes when I play games that require a lot of focus (CS, Starcraft 2) after 10-20 minutes of game I notice that my eyes are red, from not blinking enough.  If this happens to you too how do you prevent this from happening?  I know I have to blink but it's hard to remember to do that every 15 seconds when you fight with 2 armies of Protoss and Zerg.  Thanks.
Dry eye from not blinking enough
gaming.stackexchange.com
<health>
gaming.stackexchange.com/questions/18220
gaming.stackexchange.com/users/2002/Alexandru Luchian
Keep yourself properly hydrated and use glasses instead of contacts. You could also try eye drops. While many suggest to stay hydrated and use medical items such as eye drops, others suggest the seek of medical assistance.
Hydration and clean environment is recommended. The community proposes usage of glasses or eye drops. It is suggested to seek medical assistance.