Does a two pole GFCI breaker need a neutral run to the load?
Only if the load itself uses a neutral. A straight 240 volt load such as a well pump, a heater element, or a two wire spa heater needs no load neutral, and the GFCI breaker still works. A 120/240 volt load such as a spa pack with a 120 volt control board, a range, or a dryer needs its neutral, and that neutral must land on the breaker rather than the panel bar.
How the breaker senses the fault
A GFCI device compares the current going out with the current coming back. In a two pole breaker the current transformer encircles both ungrounded conductors and the load neutral terminal. If the vector sum of those conductors is not zero, current is leaving the circuit somewhere it should not, and the breaker opens. With no load neutral, the two hots alone must sum to zero, which is exactly what happens on a balanced 240 volt load.
That is why the load neutral, when one exists, has to pass through the breaker. If you land the load neutral on the panel neutral bar instead of the breaker terminal, the return current bypasses the sensing element and the breaker sees an imbalance the instant a 120 volt load turns on. That is the number one cause of a spa breaker that trips as soon as the controller powers up.
Two neutrals, two different jobs
Every GFCI breaker has a white pigtail. That pigtail is the power supply for the electronics and it always lands on the panel neutral bar. It is not the load neutral and it is not optional. The load neutral, if the circuit has one, lands on the screw terminal on the breaker body marked LOAD NEUTRAL or LOAD NEU.
Mixing those two up produces the same symptom as landing the load neutral on the bar. Section 110.3(B) of the 2023 NEC requires listed equipment to be installed in accordance with its instructions, and the wiring diagram on the breaker label is part of that listing.
Where these circuits show up
Section 680.5 requires GFCI protection for pumps and other equipment associated with permanently installed pools and spas, and 680.44 covers self contained spas and hot tubs. Section 210.8(A) drives GFCI at the receptacle locations, and 210.8(D) lists specific dwelling unit appliances. Section 422.5 pulls in a broader set of appliances and identifies the acceptable GFCI device types.
One more field rule. Section 250.24(A)(5) prohibits a grounded conductor connection to the equipment grounding conductor or enclosure on the load side of the service disconnect. If the neutral and ground are bonded together anywhere downstream, such as at a subpanel with the bonding screw left in, the GFCI will trip and no amount of rewiring at the breaker will fix it. Pull the bonding screw and separate the bars.
Field takeaway
Pigtail to the panel neutral bar, always. Load neutral to the breaker load neutral terminal, only if the load actually uses one. No load neutral on a pure 240 volt load is normal and correct. If it trips immediately, check for a neutral to ground bond downstream before you replace the breaker.