It sounds like the wire contacting the side of the fire alarm panel is either shorting out the power supply (cutting power to the door release) or making a complete electrical circuit when the wire is grounded (providing power to the release). In either case, the circuit has a second ground connection.
Most security and fire alarm power supplies aren't connected to ground on the low voltage side, but because this door release sounds like it could be for security purposes, anything goes for the power supply. If the power supply is grounded, making contact to chase with the wire would cause this effect.
If there is a door operator (the type that opens the door automatically), the door operator itself is often wired weird, so anything goes with the door operator as far as a second ground. A second ground may be normal.
If the door release is a Von Duprin type of door strike/panic bar, this has its own power supply and possibilities of wiring issues. Anyone could have wired this originally and made some weird connections. In this case, a second ground might be normal.
If it is a Von Duprin door strike/panic bar, check the wire size from the Von Duprin power supply to the door. (It needs to be 12 or 14 gauge.) Smaller than what is shown in the installation sheets supplied with the door release could be the issue. I've seen this issue show up after a year of the door release working flawlessly. The contractor then made the electrician rerun the wires with the proper size wires, which fixed the problem.
Also, if it is a panic bar, often the wires for the bar or door release may go through the door hinges (this is one of the possibilities of how the wires get to the door), and in order to run the wires through the hinge, they're very thin. There could be a break inside one of these hinges, and if one of the broken wires is grounded when it shouldn't be, that could be the second ground. In this case, the ground is not normal.
Another possibility is that there is a "Touch Type" panic bar or door handle that also electronically activates the door release. This could have an issue with the actual release circuit for the door. With these things, anything goes.
Often the whole door lock/release system is installed by three or more different vendors at many different times. None of them really know what the others has done. This scenario becomes a nightmare when trying to troubleshoot. The grounded system might or might not be normal.
When you add these possibilities to problems with the original installation, the door entry system, and the card access system (which could all be involved), without being there, it becomes very difficult to make guesses as to what could be causing the problem with grounding the wire to the chase.
Basically, fix the door release problem first. Worry about the grounding of the system later. Often fixing the release will fix the ground problem, that is if the grounded system is actually a problem.