What would happen if you just leave the wires in for your old program and u upload your new program? Will this burn the chip out in some cases? What is the best way to go about this?
As someone pointed out, it all depends on what's wired up, and what the new program does.
As far as the best way to do it, I think I REALLY like prototyping shields that come with min-solderless breadboards. I picked up some from China, on ebay; 4 of them, for about $20. They took about 3 weeks to arrive here in Canada, but they are great! If you need more holes for components, you can always add another solderless breadboard beside the one plugged into the Arduino.
Then, if you need to try out another program, you just unplug the protoyping shield, and plug it back in later, when you go back to that project. Here's a picture of one of my projects that's still in progress. As you can see, it's fairly full, but still easy to disconnect.