TPS61025DRCR regulator input voltage drops to below 3V when powered - trouble shooting techniques?
After a little more debugging, I was able to find the issue with the circuit. The following were my debugging steps:


As Peter Bennett suggested, a source voltage that 'sags' (i.e. 3V -> 1.4V) when loaded suggests that the load has high current draw, such that the source can't source enough current. The measured current was found to be 400mA - way too high. 
The next step was determining what was drawing so much current. Although I had measured and determined there were no short circuits, I did not account for active components that could become shorts under the right conditions. After analyzing the circuit, there had been a single TVS diode that had been damaged which would clamp when 3V was applied. Essentially, the short circuit only existed when adding power.


Key takeaway: if there appears to be a short in your circuit but you measure none, always keep in mind that an active component (such as a TVS diode) could be the culprit.