GFCI breaker tripping
In any installation of GFCI protected loads, they must be designed so that currents are balanced.  That is because GFCI detection compares currents in the two wires, and aims to see them equal.  As such, unhooking only the LOAD "hot" wire, and not the accompanying LOAD "neutral", is a wrongheaded thing to do generally.   However, it is an excellent and interesting diagnostic test, that tells you where your problem lies. 

What you have, by plain demonstration, is current flow on neutral that is not matched by equivalent current flow on hot.  That shouldn't be possible.  If the monogamous partner "hot" wire is disconnected, where can current possibly be coming from? 

The most likely explanation is a promiscuous neutral.  That is, this neutral is returning current for other circuits served out of other circuit breakers.  This is occurring because at some point, some installer of something decided to grab the most convenient neutral, instead of grabbing the correct partner neutral.  This often accompanies shady handyman work and overloading too-few-wire cables -- this person's situation being a classic case of both of those.

Now there's one more possibility, but it's fairly remote. That is a neutral-ground fault somewhere in the wiring.  This would most likely be coupled with hot-ground leakage somewhere else whose grounds are connected to this ground.   Current takes all available paths in proportion to their conductance, and while most of the current would return directly through that circuit's grounding system, some might hop your neutral-ground fault and return on this circuit's neutral, where it would be most unexpected by the GFCI.  Like I say, it's a longshot.  But this too is coming out of another circuit.  



Either way  -- How do you find that other circuit?  Shut off all your breakers and see if the GFCI breaker will stay on.   Then turn on one breaker at a time.  It really helps to turn on all the loads on that breaker.  When you turn on a breaker and the GFCI trips, you have found what circuit the promiscuous neutral or other-ground-fault is at.  Then you follow both circuits until you find the interaction or problem. 

Once you find the breaker, you might turn it off, reset and continue turning other breakers on - just to preclude it being three circuits interacting.  

The last thing to mention is a multi-wire branch circuit.  That is a circuit designed for two subcircuit "hots" to share a neutral.  Typically it uses /3 cable, red black white, with white being the shared neutral and red and black being hots. They must be on opposite poles or they will overload the neutral; and they need common maintenance shutoff; for those reasons we recommend using a 2-pole breaker or a listed handle tie.   If you wish to GFCI protect a milti-wire branch circuit, the only option is a 2-pole GFCI breaker. 

What about a defective GFCI?  I doubt it.  Honestly that idea is the "first place people go" because they can't believe/accept they have a ground fault, but it's rarely true.  So I tend to discount it.   You can try unhooking both LOAD hot and neutral (do not unhook the curly pig's tail LINE neutral) -- if it trips with nothing on LOAD then, yeah, it's the GFCI.