You can also have close-toed too, as well as the three you offer. You can also have the unhyphenated variants of each (a total of eight choices).
A google search suggests closed-toe is far more popular than the others, close-toe just ahead of closed-toed, and that reasonably ahead of close-toed.
I hasten to add though, that google searches are a very poor measure of these things, not least because this is not what google searches were designed for, so there's a variety of things going on that will skew the relationship between number of search results, and popularity of a phrase. It's also very bad at comparing hyphenated and unhyphenated versions (so bad, I'm not even going to try). Take the above ordering with a generous pinch of salt.
Now let's work backwards, because close vs. closed is the trickiest thing here. We'll start comparing closed-toed vs closed-toe assuming that the same applies to close toed vs close toe, close-toed vs close-toe and so on.
Closed-toed is a compound adjective, describing the toe of a shoe that is closed. Hence closed-toed shoe uses an adjective to modify a noun, signifying that the shoe has such a toe. This is a pretty straight-forward matter at this point, comparable to red apple.
Closed-toe is a compound noun, signifying (rather than describing) such a toe as we have described above. Hence closed-toe shoe uses a noun to modify a noun, signifying again that the shoe has such a toe. This is less common than the case above, but far from rare; comparable to trouser press.
We can see from the comparisons that both are correct forms. It's more common to use an adjective than a noun adjunct when one has the choice, and in the case of a simple (one-word) adjective, it would often be quite graceless to opt for stretching a noun out of its more common rôle when a perfectly good adjective will do the job. Compound adjectives are not a popular choice though, so here we'd expect closed-toe to be used by more people than closed-toed.
Both though are correct so far.
Let's now compare closed-toe with closed toe.
We would strongly favour hyphenating if the compound was likely to be part of an ambiguous phrase ("a man-eating tiger" is a threat to you, while "a man eating tiger" is a threat to endangered species). We would strongly favour not hyphenating if the compound is very common, but common is in the mind of the beholder. We might very much favour hyphenating if an eye catching the second part of the noun phrase was likely to interpret it incorrectly from just "toed shoe", but that's just tautologous and would lead that hypothetical eye to scan back again. So there's no risk.
In all, there's little to pick between them. The trend is toward not hyphenating, so we might expect that to be the more popular, though we would also expect to find some deciding otherwise.
Again, both are correct.
Finally, what about closed-toed vs. close-toed?
These use two different adjectives to modify toe, which is then in turn turned into an adjective. Closed has a sense meaning "not open, enclosed, having not openings". Close has a sense meaning "not open, enclosed, having no openings". That latter sense is old and well-established, but now pretty rare compared to closed for this meaning.
I think it's pretty certain the reason close is used much here at all, is that of sounds, such as John mentioned in the comment on the question. Indeed, I would say that of those who write it closed toed, the vast majority would pronounce it /kləʊztəʊd/ or /kloʊztoʊd/ rather than /kləʊzdtəʊd/ or /kloʊzdtoʊd/.
But whatever the reason, it still comes out as resulting in a combination that is grammatically correct and has the meaning intending.
It it didn't happen that close had this meaning, it could well still have ended up being used here since the other meanings are close enough that it may have been stretched to fill the gap.
So, of the eight different possibilities, every single one is correct. I'd favour closed toed, (but fully aware that I wasn't going to pronounce the /d/ of closed), but while you might look at some of what was said above to favour it too, any preference for another is perfectly okay.