Most of the looting -- and we know for certain there was some -- was of the "personal acquisition" category. But there are four paintings that ended up in Bermuda, which Britain was using as a base of operations.

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The British had arrived with a force 4500 strong on August 19, 1814, at Patuxent River to the southeast of Washington. The capital was clearly within reach. While General Armstrong had declared that Washington was not of sufficient strategic significance...

They certainly will not come here. What the devil will they do here? No! No! Baltimore is the place, sir. That is of so much more consequence.

...there was certainly reason for alarm even to the lead-up of August 24; Secretary of State Monroe had done some scouting with calvary and wrote a letter two days before, that "the enemy are on full march to Washington" and more ominously "You had better remove the records."

Still, even in the morning of, there was uncertainty what the British would do; while James Madison went to the front at Bladensburg where the battle went very awry, Dolley Madison at the White House meanwhile ordered for dinner to be prepared by 3:00. According to Paul Jennings (who was at the time enslaved) in his memoir A Colored Man's Reminiscences of James Madison:

I set the table myself, and brought up the ale, cider, and wine, and placed them in the coolers, as all the Cabinet and several military gentlemen and strangers were expected. While waiting, at just about 3, as Sukey, the house-servant, was lolling out of a chamber window, James Smith, a free colored man who had accompanied Mr. Madison to Bladensburg, galloped up to the house, waving his hat, and cried out, "Clear out, clear out! General Armstrong has ordered a retreat!" All then was confusion.

There are others with varying details, but the important point is: evacuation was unclear and sudden; also, according to Jennings account, before the British arrived, there was some looting:

I will here mention that although the British were expected every minute, they did not arrive for some hours; in the mean time, a rabble, taking advantage of the confusion, ran all over the White House, and stole lots of silver and whatever they could lay their hands on.

(Jennings was incidentally one of the men who helped save the famous Washington portrait. Dolley Madison gave the order to save it, but did not remove it herself as according to later accounts.)

The British -- led by Admiral George Cockburn -- arrived while the food was still set out, including, as described by a Lieutenant (later Rear-Admiral) Scott -- costly wines; Scott, by his own account, "quaffed off at Mr. Madison's expense".

One of Madison's hats made its way to the Admiral; Scott also mentions a "small portrait of the President's lady" which seems to have been absconded with; portraits of James were also stolen, as well as a dress sword. One person after dinner reportedly used the tablecloth as a sack and nicked the silverware.

Of course, there were other buildings than just the White House. The Admiral also nicked a book. We know this because in 1940 a collector gave the Library of Congress a book entitled An Account of the Receipts and Expenditures of the United States for the Year 1810 which was, as the cover declares

Taken in President's room in the Capitol, at the destruction of that building by the British, on the capture of Washington 24th August 1814

with the further provenance that it was given by Admiral Cockburn himself to "his Eldest Brother Sir James Cockburn of Langton Bart Governor of Bermuda". (You can find all the pages of the book at the Library of Congress here.)

The acquisition most resembling the original question is one that perhaps the Americans didn't want. Dr Edward Harris (of the National Museum of Bermuda) mentions four paintings found in a "warehouse". They landed on Cockburn's ship and back to Bermuda where the ended up at Bermuda's House of Assembly and the Colonial Secretariat.

You can see two of them at the House of Assembly here in a picture from 2020.

You can also see why the US has not bothered to request them back -- they're paintings of King George III and Queen Charlotte. The other pair is identical.