I'll leave a more complete answer to legal experts, but most naval shipwrecks are considered war graves, and there are very strict rules about what can be salvaged or recovered from them, if anything.

For a better answer about what could be learned, the answer is not much. World War II was not that long ago, and it's not as if we don't know "how U-boats fought" or what daily life was like on a submarine. This isn't the mystery of the Greek trireme- we have full blueprints, production records, equipment lists, operational logs for pretty much every declassified military vessel built within the last 200 years. We know the inner workings and mechanisms of a German G7e torpedo, we have physical Enigma cypher machines we can look at, and there are even several people alive who served on German submarines during the war and survived. There are even several fully intact and preserved U-boats in museums around the world. Apart from being able to figure out unexplained losses and what happened to various U-boats (which you don't even need to go inside the submarine to do), there really isn't a good reason to bother pulling one up or getting artifacts.

One more question:

Reading the following three paragraphs, I'm wondering if the U-boat was trying to surrender when they were sunk. Do any of you have any insight into that?

Here are the paragraphs:

The submerged boat closed in anyway and fired four torpedoes, sinking the old Nicaraguan flagged freighter Bluefields and damaging two other ships without causing loss of life, a NOAA account of the battle said.

Hoyt said that because of the lost weight of the four torpedoes and the broken ballast system, the U-boat couldn’t stay submerged and popped to the surface in the middle of the convoy. “They were immediately pounced on,” he said.

An armed merchant vessel opened fire with its deck gun, and two Navy escort planes swooped in and dropped more depth charges. The U-boat sank with its crew, most of whom were probably in their 20s, Hoyt said.

I want to direct you to the book Shadow Divers by Robert Kurson. It's a non-fiction account of the first U-Boat found off US shores--in 1991, off New Jersey. It's interesting from a U-Boat perspective, but also from a diving perspective. It shows the challenges and dangers of researching it (2 people died while diving there). There are 2 other books about it: The Last Dive, about the father and son who died while diving there, and what might interest you more, Shadow Divers Exposed, which is about that U-Boat and what it did during the war.

In any case, Shadow Divers explains what they were able to find on the wreck, and what they weren't able to find. Long story short: almost everything was so far degraded as to reveal virtually nothing. There are no humans remains on such vessels--they were long ago claimed by the sea. Log books, clothes, letters, all gone. Surprisingly, they also had a hell of a difficult time figuring out which sub it was--everything was so decayed that they couldn't find anything that contained the name of the sub. Eventually they did, but the dives where they searched for it were fantastically dangerous due to depth, and it was really a last ditch effort after they'd tried everything else, that led them to undertake the dive where they found the name of the sub . The divers did collect a few items of slight interest on the sub, but really, the most interesting thing about it was where it was found.

Sorry I'm late to this fascinating thread. It's the first time I'm commenting on reddit, so hope I'm doing this right.

I'm the author of the book Shadow Divers, mentioned earlier in this thread. It tells the story of a group of recreational scuba divers who found a WWII German U-boat in New Jersey waters in 1991. The sub showed cataclysmic damage from an explosion, and contained the remains of 56 German sailors inside. No one in the world - no government, navy, expert, or historian, could explain what the U-boat was doing in NJ, who was inside, or how it met its end. The divers became obsessed with these questions, and risked everything to answer them. Three men died during the exploration.

The commenters here have done a phenomenal job of discussing the issues. I have just a few observations to add. The divers who explored the NJ U-boat wreck found the remains of the German sailors inside, many well preserved, some still in uniform, their shoes lined up left-right, left-right. Often, the divers were tempted to reach into the shirt or pants pocket of a fallen sailor to pull out a pocket watch or a cigarette lighter that might be engraved with the U-boat's number. But in seeing the bones inside the wreck, and in learning about the U-boat war (crews were comprised of very, very young men, some of them teenagers), the divers found themselves connecting to the human part of the story. It was astonishing, but the divers became willing to die before they disturbed the remains of the sailors in their effort to identify the wreck. Later, the families of the German crew were deeply grateful to learn that the divers did their best to leave the U-boat as undisturbed as possible. That meant as much to them as the finding of the wreck itself.

I was surprised to learn that so much information and evidence could survive inside a sunken U-boat for so long. As it was explained to me, the stillness of the water inside, and the soot and silt that covered so much of the interior, helped preserve much of the evidence. Today, however, the wreck is far less intact than it was 20 years ago. Storms, time, and fishermen's lines continue to tear away at her.

There is an amazing U-boat on display in my hometown of Chicago - the U-505 - which is a match for the Type IXC discovered in NJ. It's really worth a trip for anyone interested in the lives and experience of the men who sailed - and hunted - U-boats. Getting to walk inside, see the machinery and the bunks and the torpedoes - makes for an unforgettable day.

This comment could possibly be deleted (and I know I'm pretty late to the thread) but there are many laws protecting wrecks, especially military ships. Referring to the Sunken Military Craft Act, any U.S. military wreckage found underwater belongs to the U.S. and the U.S. only. In addition, this applies to foreign military craft found within U.S. waters. This act protects any military vessel from the Law of Finds, as well as the Law of Salvage (unless the country who owns the vessel gives permission to salvage it). The General Conference of the United Nations Educational, Scientific and Cultural Organization (UNESCO) in 2001 acknowledged the importance of underwater cultural heritage as a huge part of the preservation of the cultural heritage of humanity, and therefore set in place many different protections for anything found underwater.

There are a lot of hoops to be jumped through before anyone can touch that site (as there are with any archaeological site). But the biggest hoop is that the U.S. respects the ownership of the vessel as belonging to Germany and Germany only, despite it being in U.S. waters. So if anyone researches it, they'll be approved through Germany first. Unfortunately I don't know what their proceedings are for these kinds of things.

Citations (do I need these?) Here's the Military Vessel Act: Sunken Military Craft And the UNESCO meeting was in Paris, in November of 2001, so you could probably find it that way.

I'm an archaeology and anthropology student by the way, and I'm currently learning about these laws in my Shipwrecks class.

Firstly, I'd like to note that though, Shadow Divers while a fantastic book and a must-read for wreck divers, is is as mentioned by /u/bettinafairchild from 1991 and is extremely dated today. In the book, diving with and decompressing using enriched air is described as exotic, while today it's used as the standard breathing gas of even introductory courses, and technical dives are routinely done down to 100m or more using trimix, not to mention advanced equipment such as rebreathers. The fatalities that occurred back then would not have happened today, at least not under those circumstances.

Now, to answer the question, it very much depends on where the wreck is, who claims it and the nationality of the divers. What you can or cannot dive, and what you may or may not salvage will depend on local laws, international maritime treatises, and you would have to be more specific about what wreck and where. To give some examples:

1. The UK is extremely prohibitive about war graves. Sunken British ships in British waters are absolutely forbidden to be visited, such as the HMS Royal Oak which was torpedoed in Scapa Flow. On the other hand, HMS Prince of Wales lies in international waters and is visited by divers, though British subjects are forbidden from doing so without permission. Like most military wrecks, both are still considered property of the British government. Also, one of the most commonly visited war wrecks in the Red Sea, the SS Thistlegorm, was lost with loss of life.
2. The MS Estonia also lies in international waters. As far as I know the owners of the ship still own the salvage rights to her, and some of the surrounding countries (Sweden, Finland and Estonia) have prohibited diving on her, considering the wreck a gravesite. This can naturally only be enforced on the citizens of the respective countries, and nothing can be doing to prohibit foreign nationals.
3. German and Japanese war graves in foreign waters can be and are commonly visited by divers, even though the modern governments still lay claim to their predecessors' ships. I have myself visited the German destroyers sunk in the Battle of Narvik, while Truk Lagoon is a commonly visited war grave in the Pacific.
4. To give an example of national laws, I can cite Swedish salvage laws, with which I'm most familiar. Wrecks younger than 100 years in Swedish territorial waters still belong to the ship owners, who can either keep or sell salvage rights (or be forced to salvage due to environmental concerns). Given salvage rights, anything can be stripped or taken from the wreck as seen fit. Wrecks 100 years or older are considered archeological sites and are thus property of the state. The state may prohibit diving on the site as they see fit (as has been done with many of the intact wooden ships which have been found in the Baltic), and may even censure disclosure of of the site's coordinates. Naturally, salvage from a protected wreck is absolutely forbidden, as is salvage of any objects found nearby (though they must be reported). Any object found while diving which can be suspected to be over 100 years of age is automatically property of the state, which may or may not reimburse the finder depending on the circumstances (or let them keep it, if it's something common).

tl;dr: It depends.

I suspect it varies; in the UK we have the "The Protection of Military Remains Act". Vessels can be designated as either "protected" or "controlled". You're allowed to dive on a protected site, but you can't interfere with it. You can't dive on a controlled site at all.

There are four German subs that are designated as either controlled or protected sites in British waters.

Of course some sites kind of protect themselves, as well as being official protected. HMS Royal Oak was fully bunkered when she sunk and she's in relatively shallow water in Scapa Flow. You wouldn't want to dive on her unless you needed to (the Royal Navy do a survey dive once a year and replace her ensign at the same time).