As there are a few different types of Invisibility cloaks, my answer will be divided into several parts.
Hair from a Demiguise can be woven together to create an inivisibility cloak. However, the hairs will, over time, became opaque thus making the cloak useless.
The information on Demiguises (or whatever the plural form would be) does not mention any restriction on their invisibility, certainly not that it would not work under water. Since knowing such restrictions would be extremely useful in finding, capturing and caring for the creatures, I find it very likely that our favorite Magizoologist would put in his book (Remember that he put tips on how to evade the law when breeding basilisks in there).
Therefore, my conclusion is that inivisibility cloaks make from Demiguise hair would work underwater.
Another type of Invisibility cloak is the one where a regular cloak is enchanted with a particularly strong disillusionment charm.
The Disillusionment Charm is described to make the subject of the spell act like a chameleon, taking on the colour and texture of whatever is behind and around them.
I see no reason why this would not apply underwater as well, and that it simply take on the colour and texture of whatever was around it.
Yes, this cloak would work under water.
The Deathly Hallow cloak
It was created by Death and managed to disguise Ignotus Peverell from Death himself for as long as he wore it. I doubt it would be overcome by something as simple as water.
Yes, this cloak would work underwater'
I am not sure you want the (in my mind) implied question of why Harry didn't use his cloak in the Triwizard Tournament answered, but I'll do so, just in case.
It is implied (by Harry when talking to Moody) that the champions are not allowed any tool other than their wands when completing the tasks (though Harry uses Gillyweed, so I'm not sure about this) and as the Deathly Hallow cloak won't answer to a Summoning charm, he couldn't use Accio to retrieve it.
Also, almost no-one knows that he possesses the cloak, but using it in the TriWiz Tournament would ensure that not only the school, but the entire Magical Britain would know about it.