I like rokpunk's first answer: no.
It's more simple than mine:
If (and that's a resounding 'if') you happen to find a FireWire-enabled mixer 'in the wild', there's a 99% chance that the FOH personnel are feeding FireWire audio to their very own DAW. Now; if you ask really nicely and bring your own blanks, you might just talk said FOH personnel out of a DVD-R of the raw WAV files so that you can play with them at home. It's not likely that they'll do it (in fact, it's extremely unlikely) but I guess it wouldn't hurt you to ask them.
As far as I know, there's no commercially-available device for passively intercepting FireWire or USB data (which is what you'd have to do to "patch in" to that feed). And your patch had best not f--k up their feed or physical assault and ejection from the venue would be the least of your worries. Anyway...
Absent a miracle of modern wire-tapping technology, getting that multi-track feed is probably out of the question unless you do it analog (which requires outboard equipment for your laptop as mentioned above). PreSonus FirePods, for instance, can be daisy-chained for up to 16 channels (but you need an analog snake or a whole lot of 1/4" phone cables to hook up to the board's line outputs). The cables/snake you use will depend on what board you're hooking to and if you're getting aux (or FX if you're crafty) sends of certain things... There are a lot of variables to consider. And you'll also need to bring an extension cord, gaffer's tape, and a power strip for your gear. Ready to get out your wallet and duffel bag?
I really don't want to discuss my (vehemently negative) opinion of multi-tracking from a FOH mixer that you're not familiar with (or currently working behind). Seriously.
A line-level feed from the main outs (to mix with your AUD source) is how I made some of my favorite recordings when I was still using PortaStudios for location recording. But I was working behind the FOH console...