And it doesn't work if you copy the files in to \some\folder\on\your\harddrive, and then do a

Z:\> mount A "\some\folder\on\your\harddrive" -t floppy

?

No. It still checks the real drive a:??

Here is the BAT file I am using.

dosbox.exe A:\LODERUNN.COM -c "mount a D:\ARCADE\CLASSICS\Lode\ -t floppy" -conf D:\ARCADE\CLASSICS\Lode\DOSBOX.CONF -fullscreen -exit

Try this:
dosbox.exe -c "mount A D:\ARCADE\CLASSICS\Lode\ -t floppy" -c "A:\LODERUNN.COM" -conf D:\ARCADE\CLASSICS\Lode\DOSBOX.CONF -fullscreen -exit

Well, now it doesn't check the real DRIVE A but neither does it recognize the disk as real. It says insert disk and press enter. I do so but nothing happens. I checked the real DISK and there do not appear to be any hidden files.

I think Sierra DISK used the same copy protection, but I don't know how that was removed. The game book says it can tell whether the files are copies or the originals. How?

One method can be to check that some specific sectors are defective.

Well, I don't know how legal it is, but I found a cracked version of the original LODE RUNNER on the web. I use the LODERUN.DAT file from the disk and the new EXE file to start the game. It runs fine now.

Most likely they used some very low-level protection like hal said
(bad/weak sectors, nonconform sector counts and so on).
Not much you can do about that from the emulation side,
besides using "fixed" files.