Short answer: Yes. Yes, they do.

Long answer: Through some unknown mechanism, the neuralyzer forcibly rearranges neural pathways, at once erasing whatever memories were there before and making the pathways plastic and receptive to new memories for a short period of time. It's specifically used to erase big, worldview-shattering things (i.e. alien encounters) that even normal, forgetting-prone brains would clamp onto like an alligator doing a death roll. However the subject's brain would naturally process and store the memories in question is irrelevant.

Also, Nume has actually been neuralyzed before, repeatedly, by his first partner, who was Not Nice and liked to use it to get her way. I think this is only referenced once in a very early Fill the Plothole story of mine, though, and the event is written so vaguely that it's probably easy to miss, so it's not surprising that it's not common knowledge.

~Neshomeh, who should get back to writing that prequel story she started this summer.

And it's kind of a long story since it involves a species that canonically has perfect memory but is still affected by a neuralyzer. That's about as simply as I can explain it, though, so... yes. Yes it can.

In the original canon they come from, neuralyzers work on anybody with the usual human eyes, or something similar. Insectile eyes apparently don't transmit the neuralyzer flash to the brain.

My guess is that somebody with a photographic memory would be affected like anyone else, but they would probably be able to detect that something had been changed. The quality of replaced memories might be more vague than they are used to; or they might remember a "time skip", a gap in memories where the neuralyzer erased them.

Photographic memory is a strange enough concept in real life. Having an extremely good memory is rare, but useful; being actually unable to forget is almost a handicap. Fictional versions of photographic memory are often in-between those two concepts--someone with such a good memory that they could memorize a page of text with a five-second glance, but not so overwhelming that they are distracted by thoughts of what they had for breakfast seven years ago.

To determine how the neuralyzer affects someone with a very good memory, or someone with a photographic memory, or someone who is totally unable to forget any experiences they have had, you would probably have to determine why that person has an unusual memory system to begin with, and how it works differently from the usual sort, and work from there.

It shouldn't matter at all. The human mind is an incredibly complicated bit of kit; rewriting someone's memories with a flash of light is completely impossible with our technology, so I'd just call it magitech and assume it does whatever you want it to.