In asoiaf, one of the crazier warriors commonly fights people one-on-one on ships, wearing armor; do we know if any vikings (or pirates, for that matter) ever wore something more cumbersome than leather armor?

Vikings wore armour: mail (AKA chainmail, etc.). Not all Vikings owned mail, but some did, and it was worn both on land and at sea.

There are plenty of examples in Viking sagas of armour being worn at sea. For example, from Olafs saga Tryggvasonar, a description of King Olaf in his ship:

King Olaf was standing in the castle of the Serpent, thus having a commanding position. It was easy to distinguish him from others, for he carried a shield overlaid with gold, and was wearing a gilded helmet, and a short red silken kirtle over his coat of mail.

He wasn't the only one wearing armour - in the sea battle the above description introduces, Olaf fights armour-wearing opponents. In a poem quoted in the saga,

The thinner of hostile ranks in the south clove with his sword hard heads of noted men. Wrathfully he passed along the sloops. The King in fight did cut to the quick the mail-covered flesh. Smitten with his sharp sword, lay many a hero of battle.

Various other pirates wore armour. For example Herodotus mentions armoured pirates:

However, not long afterwards certain Carians and Ionians who had left their country on a voyage of plunder, were carried by stress of weather to Egypt where they disembarked, all equipped in their brazen armour

Later Mediterranean pirates, such as Barbary pirates, would wear armour (usually mail).

Some East Asian pirates such as wako and Moro pirates would wear armour.

Generally, when soldiers on land wore armour, pirates would wear armour. Not all pirates, but certainly some. When soldiers on land stopped wearing armour, pirates stopped wearing armour. Armour might have been less common among pirates at sea than soldiers on land, but this might not be the best comparison - it might be better to compare the frequency of wearing armour between pirates and bandits on land. Soldiers at sea are often shown wearing armour in naval battles. For example,

As for "more cumbersome than leather armor", the idea of lightweight flexible leather armour doesn't connect to real-world armour. Rawhide was used extensively for armour in many parts of the world, but rawhide armour was thick (so as to be protective against arrows) and consequently heavy. Leather armour would need to be thicker to provide the same protection - tanning weakens the leather, and leather is not as strong as rawhide. The main exception to leather armour not being a thing is the buff coat (which doesn't stop arrows, so appears on the battlefield after guns replace bows). Vikings didn't wear buff coats (they appeared long after Vikings were gone.)

A mail shirt would often be lighter than rawhide armour. A mail shirt could be the same weight as a buff coat:

Mail shirt, 4.5kg: http://wallacelive.wallacecollection.org/eMuseumPlus?service=ExternalInterface&module=collection&objectId=60492&viewType=detailView
Buff coat, 4.8kg: https://collections.royalarmouries.org/object/rac-object-15791.html

although it should be noted that mail shirts were often heavier:

and buff coats often lighter:

If of similar weight, the mail shirt might be easier to swim in.