I need to wire a residential stairway chairlift. I have looked at Art. 620. Does anyone have advice on hidden issues with wiring?

When an issue is hidden, I don't know about it.

I hope others can tackle this.

The first thing I would do is figure out what they were installing and download the installation instructions. That will give you an idea where the listed assembly gives way to the field wiring. It may even spell out all of the code requirements on your side.

Does it just plug in? I have wired up one that only had a plug on it. I installed the receptacle where the stairs people wanted it. End of story. (I think)

The unit has a 24vdc battery. A cord comes out of the rail on the steps to a power supply that plugs into a 120v outlet.
Code calls for support and protection from physical damage...how do you support and protect a cord?

You could run it in something like Wiremold just for neatness and to keep it from getting tripped on. It would also help to keep it from getting unplugged accidently. I'd put a good quality receptacle there with decent retention force to make sure that it doesn't fall out or come loose when the rider is halfway between floors.
Probably make some notation on the OCP feeding it that it should stay turned on for the same reason.

The 120 volt outlet should be as close as possible to the unit (for physical damage protection), and the 'excess cord' should be tie-wrapped to prevent any trip hazzard.

There are cord 'protectors' on the market, but you should check with your local AHJ if they are acceptanle in his opinion.

Using 'wiremold' (surface metal raceway), IMHO is not code compliant.

For support....the unit should have a cord connector (strain relief) from the factory; straping, stapling etc. would be considered using cord for a permanent wiring method which is not code compliant.

While I agree that the permanent installation of flexible cord in a raceway is generally not permitted (400.8(6)), this is a power cord of an assembly that only needs to be protected between the outlet and the device.

I see many wall-mounted flat screen TVs using a Wiremold-looking product to neaten up the appearance of the installation; I think that this is something that the AHJ should be asked about. He may grant acceptance of it in light of reducing a greater (tripping) hazard.

http://www.legrand.us/Wiremold/At-Home/Hide-Organize-Cords/C100-C110-CordMate-Cord-Channel.aspx is one of a number of similar products that purport to address this issue but I don't know if they are UL Listed.

It sounds like a wall wart that keeps a 24v battery charged (probably two 12v Gel Cells) The "cord" is low voltage.
If this is the non-metallic wiremold or wiremold cord protector product I don't really see a problem using a short piece to cover and protect the cord. Just be sure the edges are deburred and radiused so they don't damage the cord. Metal surface raceways without the proper terminating boxes do pose a number of problems.

Ghost:
Your comments above are understood, and yes I agree that this type of product is useful for minimal protection. Yes, it is neat.

All that said, and something that I should have mentioned previously, the described install responsibility for the EC IMHO ends at the receptacle the EC installed, along with the branch circuit and OCP. A 'plug in' item is not within the responsibility of the EC.

BTW, I see no indication of any UL/CSA at the links, but verbage indicating not for use to extend permanent circuits.