I've birthed standing up twice.
First time was in a shower with no tub, but with a seat. I had my right leg up on the seat, my left leg on the ground. I had my back to the MW and DH and was leaning into the wall of the shower. 10 minutes of pushing, and I had a small tear that I had 1 stitch in, although the stitch wasn't necessary. When DS was born, I turned around, grabbed him, and sat down on the seat in the shower. About 10 minutes later I walked over to the bed (with help). I delivered the placenta in bed about 30 minutes later.
Second time was in a shower with a tub (at home). I was standing with both feet on the ground. I was leaning over with my hands on the side of the bathtub, facing DH. No tears that time, about 10 minutes or so of pushing again. After DD was born, I stood in the shower while DH made a little spot for me in the bedroom. Then, he helped me in there to lay down and nurse DD. After snuggling and nursing for about an hour, I was getting really uncomfortable with the cord and placenta, so we cut the cord, and I went back to the shower to birth the placenta. Took about 5 minutes. Then, I washed up and went back to snuggling DD :-)
The idea about standing / tearing is that standing makes the birth happen faster. The faster the birth, the less time your tissues have to stretch, so the more likely you are to tear. While this is logical in theory, I don't know if it is true or if there is an scientific evidence that says it's one way or the other. I've had two births with no tearing at all - one on my back, one standing. The laying one was faster - one push and she was out - and still no tearing. Of course that's just anecdotal. If you are worried about tearing, I would do some research on the subject to see if there are any scientific studies or information beyond anecdotal.
I think standing births rock, but so do water births, and I haven't yet figured out how to be submerged in water but still standing, at least in my home :-)
Oh, and like PP said, there was no reason I couldn't keep standing after those births, but I was on the birth high and tingly, and I wanted to lay down to nurse and focus on my sweet babes.