The irony of the Chestnut Cafe is simply this: in a bar, people are at their most relaxed. Patrons go to a bar to do what they want to do, drink, play pool, meet friends. The whole concept of a meeting place, bar, relaxed atmosphere is mocked because the Chestnut Cafe is a completely controlled environment pretending to be something that is no longer part of society, a place to be free.
Winston is drawn to an environment that should provide relief from daily life, but the Chestnut Cafe is just as mechanical as the other aspects of Oceania. It is trap where you are lured under false pretenses only to be subject to more indoctrination.
"A waiter, again unbidden, brought the chessboard and the current issue of The Times, with the page turned down at the chess problem. Then, seeing that Winston's glass was empty, he brought the gin bottle and filled it. There was no need to give orders. They knew his habits. The chessboard was always waiting for him, his corner table was always reserved; even when the place was full he had it to himself, since nobody cared to be seen sitting too close to him. He never even bothered to count his drinks." (Orwell, Chapter 6)
Winston's fun is as meaningless as everything else.
The irony of the Chestnut Tree Cafe is that although all of Winston's needs are met there and every effort is made to ensure his physical comfort, none of it matters anymore. Like the Victory Gin which is made with artificial sweeteners to mask the horrible stink underneath, the Cafe is a facade, camoflaging under the guise of luxury and civility the fact that Winston is essentially dead. Since his release from the Ministry of Love, Winston has been given "a sinecure", a job in which he is more highly paid than he has ever been. He has plenty of money now, but spends his free time at the Cafe in a drunken stupor. The waiter, knowing his habits, supplies him with an endless supply of gin, and, unbidden, brings him the paper and his chess game as well. This attention means nothing to Winston however, because by betraying Julia, he has committed the ultimate degradation, giving up the last shred of his dignity and integrity. Winston will be exterminated soon but he doesn't even care, because all that is important in him, his very humanity, has been extinguished already.