The game seems to let you swim as deep as you want. Are there ever any items to be collected? Or any danger under the surface of the water (like sharks) ?
Yes, there are collectibles underwater - mainly black boxes that you need to gather for the Reapers. Look for the blue "blips" on your map and you will see that some of them appear in the ocean.
I've yet to encounter any dangers underwater, though, so I don't think there are any sharks or the sort.
As Ragnar said - black boxes. I don't think I've found anything else under water. Most of the boxes are in the shallow areas, and none of them are very deep.
The only danger underwater is running out of oxygen - but this takes a surprisingly long time to do. As a result, diving turns out to be a good way to shake off excessive heat. You don't have to get down too deep before even the helis either can't see you, or their bullets can penetrate, either way, safe. Since the oxygen burn rate doesn't seem to depend on activity, you can swim a significant distance before you surface, and that help shake heat, too. So - dive, swim on some heading until (1) heat's gone (2) you've healed enough or (3) you're down to 20% O2, then come back up.