You wouldn't bother.
Immersion (wiki) :
The state of consciousness where an immersant's awareness of physical
self is diminished or lost by being surrounded in an engrossing total
This term is often used in the wrong way ; people think that good graphics makes a game more immersive, or that a real physics simulation will do the same.
This is wrong, the biggest immersion breaker is incoherences in the world in which the player is. Like the physics engine messing up and sending random things flying, or a stupidly low res NPC. Something that is really immersion breaking (at least for me) is game rules being broken or frustration.
Once the player has made a conscious descision to join/leave the world they accept that they are entering/leaving the game world. Not breaking immersion would be solving any disconnection problems and, failing that, make it less frustrating for the player so they can pop straight back into the action. You could also consider that those who are playing with a player that gets d/c'd have their immersion broken because the guy just dissapears.
Of course the immersion of those staying in the game is momentarily 'fissured' because someone just dissapears. But even if you found some fancy animation, people would still know that the player was logging out. Some people may even get a chat message to say their friend has left the game - logging out is just an accepted part of the game.