MSP4302274 Custom board, code uploads but nothing happens

Sure. The node labeled 'SRC_MSP' has been jumped directly to 'VCC' on the actual hardware. 

RESET pin must be connected to VCC over 47 kohm resistor and to ground over 1 nF capacitor. On your schematic RESET is not pulled up, so device after disconnecting from FET will never start with program execution.

For checking if device was flashed correctly, can be used any type of verify command.

Hmm interesting. This is the schematic from the actual development board. Could the schematic be incorrect? And actually, on my schematic the 47k is connected to the reset and VCC (which is labeled SRC_MSP).

Dev board schematic is correct. There is 47 kohm pull up on RESET line. 2 x 100 nF caps are for VCC decoupling, not related to RESET. 2xx device can run without 1 nF cap on RESET pin.

Don't see connection point on wire crossing (SRC_MSP / RESET) in your schematic, so 47 kohm resistor together with cap is connected to ground.

For some reason my schematic editor didn't create the node connection marker at that spot, but indeed they actually are connected. I've verified this on the layout as well. Sorry to not have been clear about this. 

Any other thoughts as to why this could be happening? Are there any instructions I need to load to the chip before code? I read in the datasheet the USB bootloader is pre-loaded. 

Interestingly it was only a section of the pins that would not respond to the code. I swapped out the chip and it all worked fine!