I am making some TIG electrode holders out of 1/2" copper pipe. I had some pipe sitting around after taking out a bathroom in my home that was inexpertly and in-advisably installed, so I figured I had nothing to lose. I don't have a torch to sweat them, so I figured what the heck and I lit up on them to see if I could weld them. I had read a lot about how copper soaks up the heat and requires a lot of amps, but this stuff is so thin I figured it was worth a try.
I started at 160 amps and promptly blew a hole right through the pipe. I guess I have enough heat! Those who have worked with conductive metals like aluminum or copper won't be surprised to hear that there was a huge difference in behavior between when the metal was hot and when it was cold. When it was cold, I couldn't even get a puddle going at 100 amps, and then as soon as the puddle formed, it was huge and wanted to blow through. Once the metal heated up, though, around 90-100 amps was just right, for tacking anyway. I didn't have any filler, and I did want to produce a few usable pieces, so I played it safe and just tacked up. The torch switch, with fixed output controlled by the panel dial was perfect for this. If I had been using the pedal, it would have been very easy to either overheat the piece with too low an output, or overshoot and blow through. With the torch switch, I could hit the exact amperage required immediately. I suspect, however, that if I had been doing more than tacking, I would have really wanted the pedal's ability to back off the heat, because the behavior of the piece really changed a lot as it soaked up heat.
Anyway, the answer: yes! You probably can weld copper pipe with your TIG welder! Here are some pics: