What is the green sediment in the ice cubes from my refrigerator ice maker? Is it
harmful/

Many icemakers have copper pipes. When copper oxidizes, it turns green.

Another possibility is algae.

I expect that it is algae.  Has the fridge been disconnect for a while?
Ask the maker or a repairman how to clean the ice-cube system.

The green sediment in your freezer is algae. I have had it in an old
freezer i keep in my basement at university.
Answering your question, Algae is not harmful to Humans (although if
you are fish, beware!). In some cultures algae is used in cooking, for
example, Chinese 'Fat Choy' (cyanobacterium, or, Blue-Green algae),
and also in Vitamin C supplements.

However, to remove this, defrost your freezer and scrub the insides thoroughly.

Hope this answers your question.

(This is assuming that you get your water from somewhere other than
your own land, like, you're on city water, or well water, or something
normal, not a kiddy pool piped straight into your house)

This is not algae. Unless your water supply is so horrible you have
algae oozing out of all of your faucets and you're on some messed up
water supply ran by Billy Bob.

Algae requires light. You cannot have algae grow in the dark. It's
impossible. Plants need light. You're not going to find algae in well
water, or city water. Chlorine is added to your water supply to
inhibit the growth of anything.

So the only chance of this being algae, is if your water sits,
unchlorinated, uncovered, in the light somewhere for a LONG time (1-2
weeks). No water supplier can legally offer water to your house like
this.

It is corrosion. This is especially common if you have water with a PH
lower than 7. Many cities with acidic water add carbonates to raise
and buffer the PH to eliminate this type of corrosion.

Don't believe me? Go to Walmart, or wherever, and get one of those SPA
PH test kits. They're not expensive.

I used to have a trumpet, and I'd get nasty green sludge that would
come out of that, from the corrosion of the copper in it. (Well,
combined with Diatmons or something else as well, but the green is
definetly copper).

Things that grow in water:

Algae (Green, needs light), Diatmons (Brown, does not need light,
encountered with high silicates in the water), Fungus (Can be green,
unlikely, occurs when you have high nutrient levels, like a dead
animal rotting in your water pipe, not likely), Bacteria
(Cyanobacteria, greenish/blue, unlikely, needs light).

If you don't want to spend the money for a water test, go to petsmart
or petco, and take a water sample. They will tell you what your PH is
(IF it's less than 7, then corrosion is your problem). Also they will
test for chlorine (it evaporates out, so you might not have any by the
time it gets to them).

The algae's only possible if you had a ton of water sitting in your
fridge, and left it out in the sun. If it was this bad, you'd also
have mosquito's breeding in there.

Algae's not bad for you. If you have other bugs in your water, those could be bad. 

Large amounts of copper might be toxic if you have a very bad problem.
Most metals in excess are toxic.

Small amounts of copper in water are extremely toxic to snails and shrimp.

Many common 'snail killers', use copper.