Okay, that explains the burned fingers. BTDT.
I use long straight pins. Basically you have to nip a little of the ironing board cover, expose enough of the pin so that the folded bias tape will fit under it, and nip a little more of the ironing board so the pin remains secure. Place your iron so you know where to place the second pin. You do have to iron the first foot or so that comes out of the bias tape maker with the finger-burning technique; however, once you have enough bias tape, you just thread it under the two pins, place your iron on top, and pull the bias tape underneath the iron; it will be coming under the iron right out of the bias tape maker. You can lift up your iron if necessary.
Assuming your bias strips have been sewn together, you can make yards of bias tape this way very easily. The hardest part is getting that first foot of tape ironed. I just make sure to pull the strip through steadily -- not so fast that anything is going to get out of shape, but not so slow that I risk burning the fabric.
Oh. I should mention that I also heavily starch the fabric before cutting it into bias strips. However, you don't have to do this. If unstarched, I would probably be lifting up the iron more to make sure I wouldn't be stretching the bias out of shape; might work without doing that, though.