First thing to do is verify that it is overheating. If it is, that could cause crashes and video corruption.
Best way to check is to use an application that will monitor the temps realtime. I use the CPUID Hardware Monitor. The price is right (as its FREE!!!)
http://www.cpuid.com/hwmonitor.php
Install and run it - then run your games. What you want to look at are the third and fourth sections. The third is your CPU temperatures (plural since you likely have at least a dual core cpu) - the fourth is your graphics card temperature.
Heat wise - you don't want your CPU to ever hit above 60 degrees C.
The GPU (Graphics card) per ATI can go as high as 104 degrees C. Preferably, you want it at 95 or under, but max is 104.
Run your game for a while, and every once in a while - Alt Tab out and check the temps. If you see them climbing into dangerous territory (its ok if it heats up - just it should stabilize), then you know that whatever your playing is stressing the component that is heating up too much.
Heat buildup can come from any number of things - but in most PC's they end up almost always being due to crappy air flow. So first off - where is the PC box itself? Is it in one of those "hideaway" computer compartments under a desk? Places like that are enclosed - meaning that even if its pusing air well - its working with hot air so its going to heat up. (My rig has to come out of such a cubby and have the side off due to air flow issues - I am dealing with it only because I will be switching to liquid cooling soon.) Make sure it has plenty of room around it to get air. Also make sure all the fans are turning in the case, and see if you have spots in the case for additional fans to help suck in cooler air and exhaust the hot stuff.
If you are overheating - take the pc back to the shop - and show them how you overheat (make em run the game or whatever of 10 minutes or so and tell em what temps you were seeing) - let em know that they have to keep the box cooler - but only do this if you know that where you keep the box has plenty of airflow. If you keep it in an enclosed spot and it heats up, pull it out and play it awhile just sitting in the open - see if that solves the issue first. If so, fans won't help - you gotta give it air. If it overheats in the open - then they need additional fans to keep it cool.
Of course - if overheating isn't the issue, then all this is moot. However, the fact they can't find a problem - and the corruption / crashing isn't constant - but is there in gaming - reinforces the heat possibility.
Let us know if there is more we can do. Good luck.