The idea with modeling gear is that it is supplying ALL the character to the tone by itself. So, if you put anything after it in the signal with its own "character" you're going to be altering the signal away from what the modeling gear is trying to accomplish. Hence why people have urged you to not run them through a tube amp. You absolutely can... it's just not going to give you the best representation of whatever you're trying to emulate. Example - If you're trying to run a model of a Vox AC30 through a Fender Twin Reverb... you're missing the point. haha.
I had a PODxt Live for several years, but have since sold it due to lack of use... When I did play with it in live shows, I plugged straight into the house sound system of the venue I was at or straight into a mixing board of a PA system. When practicing, I had a little solid state practice amp I ran it through. When recording, I plugged straight into my preamp/recording interface.
I'm sure that most of these guys are going DI into their interface. They can chime in if I'm wrong.