If lounge chair is taken away, can you still play your game?

Yes, the storyline and limited-time events still continue, just no lounge or team activities.

The lounge chair should not be missing. This is a glitch. Try restarting the game after you restart your phone. If this doesn't help, you better contact Wooga support using the help button in the game. I guess you can continue to pay without the chair but you won't be able to exchange drinks which means you won't be able to complete some Help Wanted tasks and that will prevent you from playing a regular game.