No, there won't be a problem at all. In fact, your doubts and the first answers are something very weird to say, in 2015. You bought a computer in order to use it as you please. Gaming on it 24/7/365/10years won't do any harm to it, unless it had some defects. If, for some reason, it reaches some critical state, the computer will shut itself until it cools off.
I have a 2011 Air and for months I used it regally during the day and let it convert videos (always max CPU usage) during the night. The computer was built to never pass 105 degrees, so there isn't any problem. People talking about cooling pads and so on have no idea of what they are talking about. It's a waste of money.
Now, you bought a Mac and not a crappy PC. Your Mac is made of an aluminium-based alloy, not only because it makes it look damn cool, but because it conducts heat. So the chassis does the same as a fan. That's why no one in their right mind should ever invest on plastic PCs. Alloys like this were expensive, but Apple and Apple alone made it possible to become mainstream.
So, a Mac will always seem "hotter" than a regular plastic PC because it is sending heat through the chassis, while in the PC the heat is trapped inside. That is specially dangerous to computers with AMD/NVIDIA GPUs. So, this means that no matter what some people want to say, if you want a long lasting PC for play any sort of game or doing anything minimally intensive, it has to:
a) Be built with the same alloy as Macs, with similar-copied thermodynamic solutions;
b) Be very, very big and thick with many fans.
In the first case, you have computers like the Razr Blade, copying as much as they can from the rMBP. Take this in consideration: Apple's approach is so superior, that they are the only company building 28W Intel chips on computers with 13" (or less) screens, and in order to fit a discrete graphics card, Razr builds a computer close to the 15" rMBP, and can't price it as cheap as the 13" rMBP because they aren't as good with design and engineering, supply chain and economies of scale. As such, they don't sell and no one gives a f***. Heck, no one of does rMBP clones sell in any meaningful quantity, and they just give problems, or are as expensive as Macs. (If you have the money, no one chooses a regular PC over a Mac).
In the second case, you have monstrosities like Alienware's machines and some Asus Machines, that no one also cares about as well. Call it enthusiast's machines. A small, irrelevant but vocal niche.
So yeah, play as much as you can, torture it, and enjoy it. My MBAir is amazing and I won't ever buy another computer from another company, as long as Apple keeps making such amazing products. Now, that TN panel was long in the tooth when I bought my MBAir...