Ok, I have been over and over the tutorial. I can't find anything wrong. I tried 2 different brands of batteries, both heat up so much they melted the housing and burned my fingers pretty bad. Nothing on the board itself heats up at all though. Not a single piece heats up.
So why are the batteries melting the case (2 of the batteries literally melted themselves too)? Is there something not installed correctly? Defective piece?
Sounds very much like the batteries are short-circuited. From the photograph, it looks like the leads on the solder side haven't been trimmed very short. If any of the soldered leads are in contact with metal while you're doing your testing, then it could explain the symptoms.
Otherwise, from the pictures, I can't really see anything wrong with the assembly. With your multimeter, check for continuity between the red and black battery wires - there shouldn't be any.
If you have a multimeter, measure the resistance between the red & black wires where connected to the PCB. If it is zero - or very close - you have a short somewhere.
If there is a short. Disconnect one of the wires from the PCB and measure the resistance between the wires again. If it still measures zero, the short is in the battery case. Otherwise, it is somewhere on the PCB.
0.2 ohms would indicate a short circuit. The next thing to do is disconnect one of the battery leads and test again, as suggested by adafruit_support. That wil tell you if the problem is in the board or in the battery holder.
OK, then the short is on the board someplace. Touch up any solders that look rough or frosted. Trim the component leads short on the solder side, and examine everything carefully for any bridged solder connections. Check on the component side as well - sometimes excess solder gets sucked up through a hole and causes a bridge on the component side.