I have read here (and many other places) that connecting pins 5 and 6 will wake a Pi from sleep/standby. I have a few pi's that will be shutting down remotely at the same time. I would like to connect all of their pin 5's to one side of a push button switch, and all of their pin 6's to the other side, so that pushing that button will connect all of their pin 5's to their pin 6's. Is there any danger to the Pi's from having their pins connected in this manner?
You can (and indeed SHOULD) connect the Gnd (pin 6) together if the Pi are to be interconnected for any reason.
Connecting pin 5 (SCL) of multiple Pi is a BAD idea. The pin is used for many purposes, usually output, and connecting multiple outputs together is always a bad idea, compounded in this case because each Pi has a 1.8kΩ pullup.
If you really wanted to do this you could isolate each Pi with a diode, which would allow the pin to be pulled low without interfering with normal operation.
NOTE the tutorial you linked is obsolete, and recent Raspbian have shutdown functionality inbuilt
dtoverlay=gpio-shutdown , so NO code is required. See Raspberry Pi 3 model B "dtoverlay=gpio-shutdown"
Also all the modern Pi have
Run pins (which are actually Reset pins), which are a far better candidate for restarting - but this is independent of shutdown function.