Even with the PCB schematic and/or layout (are 2 PCBs in play?), and part numbers for the DAC, level shifter, and other parts... this would still be quite a feat to remotely (no hardware in hand to test) troubleshoot over the internet.
Some questions on the diagram...
I see a block called "Level Shifter", with 3 power wires and 3 signals from Teensy. But it lacks output signals. Also has no connection to GND. I'm guessing the actual connections are more involved that we get to see in this diagram?
You mentioned the DAC is on another PCB which is connected by a cable. In the diagram I can see 3 power wires, and 3 pins called "SPI Pins" which don't seem to connect to anything. I can't see any indication of connectors or a cable.
Most DAC chips require at least a decoupling capacitor. Some take other parts. Nothing extra is shown in the diagram.
Still just wondering what the DAC part number is, and which level shifter is used. Can't see any part numbers.
But without this info of the hardware you're connecting, I can at least guess about the heating issue. It's very likely coming from U4, the 3.3V regulator.
Two large vias on the sides of U4 allow its heat to travel into the ground plane inside the PCB, so when U4 dissipates head that entire area of the board tends to become hot. It can be difficult without thermal camera to tell exactly which part is generating the heat, but U4 is the most likely.
If it really is U4, that's a strong sign something is drawing a lot of extra current from the 3.3V power line. From your diagram, sounds like the DAC doesn't even get a connection to 3.3V power. Very difficult to imagine how this is happening, based only on the info shown. By guess is the actual hardware is more complex than we can see...