Do you need to let a refrigerator settle after moving?

Generally, no. There is nothing that should be upset by being jostled. I would first like to say that I have no empirical data to back this up, and my answer is strictly educated conjecture based off personal experience.

We have a mini fridge that uses a "floating bearing" design. It's in a sealed can so I can't crack it open to poke at it, but my guess is the moving component of the compressor is suspended in the magnetic field when running. It doesn't like being jostled, and makes nasty noises if you do it while running. While trying to start it off an underpowered inverter (redneck camping), it would move into a position where it wouldn't run correctly. After a few minutes it was fine and would start. If you plug yours in and it doesn't sound right, i.e. quieter than usual, but more of a hum than a whir, unplug it and wait a minute minute before trying again. I wasn't able to produce this by mechanical means, so it may not apply.

If you moved a refrigerator on it's side, put it upright and do not plug it in for AT LEAST 24 hours. Refrigerators use oil in the freon to a lubricate parts of the compressor. If you laid it on it's side to move it, the lubricant flowed to the cooling lines and if you stand it up and plug it in right away there will be no lubricant in the compressor and it will fail within a few moments.

Stand it up for at LEAST 24 hours to give the lube time to flow back to the compressor.