Thanks! I tested it in Mesen and unfortunately it doesn't appear to simulate the disk access LED on the disk drive, so there was no way to know if it accesses the disk for sure. But it does pause with a black screen at some points so I'm pretty sure it's accessing the disk at those points to read or write to it.
I should just explain how the disk system works: The BIOS reads data from the disk to the RAM adapter at boot (files to be loaded by the BIOS is specified in the game's disk header). After that the game is fully playable like a normal cartridge game since it's running on the RAM adapter (which works just like a normal game cartridge as far as the Famicom is concerned). The disk drive doesn't use any power at this point (which is why batteries lasts so long) and you could even unplug the RAM adapter. As soon as it needs to access the disk (either for reading to load more data or writing to the disk to save data) it must be plugged in and powered though. When it does, the red LED lights up on the disk drive.
With my observations using this hacked disk image in Mesen, the number of points of gameplay it needs to access the disk, not counting the boot load, is 5 (although it may vary depending on how you count):
1) After saving the Toad in stage 4-4, to load worlds 5 to 8.
2) After reaching the door at the end of stage 8-4 and D-4, to load the
ending.
3) After the ending, to write a star (8-4 ending only) and to load world 9
(8-4 ending only and only if all Toads are saved).
4) After the ending of 8-4 (only if not all Toads saved), after the ending
of D-4 and after a miss in world 9. In all cases to load title screen and
worlds 1 to 4.
5) When pressing A+START to enter the letter worlds (requires 8 stars).