I am remodeling my house and have run a variety of colored wires through conduit to a junction box. One branch circuit from that run goes to outdoor outlets and another goes to a double gang outlet for the entertainment center.

Now that the entertainment center outlets are being used, the GFCI breaker trips when anything is plugged into the entertainment center.

Is this caused by sharing the neutral between the two circuits?
Can the neutral be shared between a GFCI or Arc Fault breaker and a standard circuit?

Yes, you cannot have a shared neutral on a GFCI. In a shared neutral, the neutral, acts as a return line, per se, and when one circuit is used, it senses an imbalance and trips. Why not leave the outside protected and pigtail off before the GFCI?

Mr. Wizard...
Initially I'm concerned that you're sharing a neutral... yes, it is an approved wiring method but in many past projects has caused many failures. As a standard company policy we have forbidden this wiring method from all of our projects. What have you gained or saved? What is the risk?
My recommendation to you would be not to share any neutrals, separate your circuits...
I strongly recommend getting a permit and an inspection, most inspectors I know are more than happy to help with DIY projects.

Sharing a neutral with a GFCI breaker will never work as the current through the phase conductor and neutral will often be different which is exactly what the GFCI is designed to protect against.
Sharing a neutral with an AFCI might be OK but does it violate the installation instructions?

Keep your projects simple, don't cut corners and follow the listing and labeling instructions for everything you install. Good luck

Thank you, Great answer and advice.
The reason in one case was a 4-way light circuit was wired with one wire shy to properly wire the three switches. One solution was to install hi-tech expensive Lutron (or equivalent), and the other was to borrow a neutral from a nearby outlet on a Arc-fault breaker. As you can guess, the arc-fault tripped as soon as the lights were turned on.

In the GFCI case, exterior outlets were added to a junction box under the house that contained wiring for multiple circuits. Several neutrals were tied together and the neutral for the new outlets was added to the wire nut. I have since isolated one neutral to dedicate to the outdoor outlets and all is well.
I have learned a lot from this project, and 3-way/4/way light switches are now second nature.
Inspections are forthcoming and I have a good working relationship with the inspector.

Yes, you cannot have a shared neutral on a GFCI. In a shared neutral, the neutral, acts as a return line, per se, and when one circuit is used, it senses an imbalance and trips. Why not leave the outside protected and pigtail off before the GFCI?

This was a mistake made within a junction box. I wired to a neutral that went to a dedicated circuit for the entertainment center and the problem wasn't discovered until the first time I plugged in a computer to the outlet.
I wasted time troubleshooting the outlets in avoidance of the uncomfortable crawl under the house (I thought I was done ;-) )
This has been an ambitious project
Thank You

Wirewizard...
Excellent... The more you learn the more you discover there is to learn.
I'm pleased to hear you are having the local inspector review your work,
Many shy away from permits and inspections when the inspector is another
Resource for support. When many permits only cost $50, why not?

The 4 PVC conduits passing through framing members entering panel will need nail plate protectos, same as over the low voltage wiring.

I do not see the neutral marked white, if those are the 120/240 volt feeders at the bottom of the panel.

Romex to right of CCTV box not stapled properly.

At the 4 gang, inspectors typically want to see sheathing removed, and at least all gorunds splices and pigtailed ready for devices.

All of those issues have been addressed. Nail plates have been installed even where they weren't needed.
This was early pictures of the work I did, part of the photo portfolio to show what is behind the walls.
This house is fully dry-walled and painted at this point. Appliances and A/C to be installed soon.
Thanks for the advice, I am way past minimum code requirements ;-)

Wirewizard...
Excellent...The more you learn the more you discover there is to learn.
I'm pleased to hear you are having the local inspector review your work,
many shy away from permits and inspections when the inspector is another
resource for support. When many permits only cost $50, why not?

I agree. I wired the low voltage for a friend who was remodeling a small office. At my insistence he removed all his romex boxes and installed smurf and increased his interior wall depth to accommodate the conduit boxes. His argument: I'm not pulling a permit, no inspections.
After following my advice he was "discovered" by an inspector who was so impressed by the work there was no fine imposed and the work was allowed to continue. A great lesson, do the right thing even when nobody is looking.
The photo attached is above the ceiling in a lawyer's office.