I can not directly answer when it was first used in naval warfare I can help you narrow down the time periods somewhat. We can say for certain that it was around in the early 17th century. The ill fated swedish warship Vasa carried rudimentary chain and bar shot when she sank in 1628. We can also essentially rule out any use prior to 1400, stone shot or lead covered stone shot were the norm before that, the casting techniques to produce chain or bar shot were simply not available prior to that. This is still a 228 year gap but it's a start.

It is worth noting that the chain shot on Vasa was relatively rudimentary. The chain had only 3 links (two longer oval links and one short round link) and was not hollow. It is in essence just two half-balls joined by a short chain.

I'm going to get a bit speculative now but Sweden was one of the major foundry nations in Europe from about 1640 to 1750. Cannons such as Blackbeards massive 1350 lb beasts on the Queen Annes revenge were Swedish made. If the chain shot carried on the Vasa was rudimentary, it may be reasonable to assume that we would not find much more sophisticated chain shot elsewhere at the time. I would therefore guess, based on these two pieces of information (Swedish foundry expertice in the early 17th century and the rudimentary chain and bar shot found) that chain shot was still being developed at the time, placing its age some time in the early 17th or late 16th century.

As to when it was first used and how effective it was I'm hoping someone with more naval expertice can pitch in. Chain shot is inherently less accurate than ball shot and I've seen some references to it being difficult to aim for the mast at anything beyond 100 yds but nothing conclusive.

Artillery through the ages - A. Manucy
Vasa - F. Hocker

Alright, this is something I've had asked to me before.

For the English, the use of chain shot, bar shot, double-headed hammered shot, hammered shot, and cross bar shot possibly have their origins possibly back to the times of the Mary Rose, definitely by first quarter of the 17th century. According to Margaret Rule, The Mary Rose: The Excavation and Raising of Henry VIII's Flagship there was chain shot. But, see my last paragraph here for my concerns on that.

If you are wondering what the difference between all the above different shots are, double headed shot meant a shot with a bar that had two spheres at the ends, hammered shot meant a bar with cylinders at each end, and bar shot meant literally just a bar with nothing at the ends and wooden sabots around it to make it fit a bore. Some variations in name also came from how the shots were made, such as by forge or by casting.

Oh, and let's not forget the chain shot in question, which literally is two spheres connected by chain. It's hard to nail down it's first use since the English Navy may have lumped chain shot with double-headed shot in gunners' stores records. Meanwhile other naval documents, naval dictionaries, or even regular dictionaries of the 17th and 18th century had the chain shot showing up regularly by the late 17th century. It would make sense since the late 16th and early 17th century is when Naval warfare began to adapt to more regular use of ships as bearer of guns - at some point more different kinds of shot would be tried for different purposes. Sometimes one needs to disable the sails and rigging of the opponent. While a shot to the mast with a round shot would be effective too, there's also the good chance you'll miss. Why not try and design shot that did a better job of destroying rigging - especially as range became better (and guns on ships became useful for more than just targeting crew or things that may give cover to crew/enemy guns).

How effective was it? It appeared to have some effect since it has continued use through the Age of Sail and keeps on showing up in the archaeological findings on wrecks. Chain shot in particular had a advantage in being able to spread out more than those connected by bar. But, I do suspect that, at least in terms of the Navy, maybe ones connected by bar are talked of more or maybe even used more since you can't cast a chain shot - it has to be forged. Forging a chain to make the chain shot and connect them to the two shot is a lot of work - and is all done by hand (every single link is made by hand). a double-headed shot could be cast, or at least forging it would take less work. If you want to see the effects of chain shot, there are several television shows from education specials that have replicated this shot (though, if I remember correctly, the "Warrior Graveyard" episode "Navy of the Damned" did the best job - if I'm remembering my shows correctly - that show is on Neflix if you want to see it).

My overall source is Brian Lavery's Arming and Fitting of English Ships of War, 1600-1815.

Now, for the person who asked this question - check out or buy the books Before the Mast: Life and Death Aboard the Mary Rose, Weapons of Warre: The Ordnance of the Mary Rose, and Sealed by Time: The Loss and Recovery of the Mary Rose since all three of these are part of a series on the archaeology on the Mary Rose and it's history. I've obtained quite a bit of information from the first book listed above. The second one will be the ultimate answer to your question (for one it's 2011, Lavery's book that covers a larger period of time was from 1987, the newer book may present new findings not previously mentioned by Lavery) for chain shot on the Rose. It's more up to date than Rule's book on the Mary Rose that came out in the 80s. What I'm wondering about is context. If there are chain shots on the Mary Rose, I'm wondering if they are literally some of the first for naval use. Were they just trying them out? I suspect they aren't a large part of the armament on the ship. It's all about context, and that later book on the subject would have better answers (I, unfortunately, don't own that book).

Hope that helps answer your questions.

EDIT: For shot on the Mary Rose the big story these days is the find of a lead cannon ball with iron as the core - a prototype piercing round some refer to it as. Read about it here. Like I said, the Mary Rose might have been carrying a variety of shots just to see if they had a chance to use them and see how they worked.