When I got all my carpets fitted (the whole house) in one day the fitters were quite obliging. They let us clear one room at a time and helped move furniture around. I think most fitters are fine about this, but if you're not sure call the company providing the fitters and double check. plus good tip stand on guard with the hoover because you'll want to hoover the mess before putting the furniture back!!
I think they expect to have all the rooms clear, but may offer to help you do it for a fee.
When we had most of our house done, we had it done over two days, so that we could move the furniture around. It still wasn't easy. They tend to charge by the room, so it doesn't cost extra for two visits.
Carpet fitters work very fast, and you wouldn't be able to shuttle the furniture between rooms in the time it takes them to do one room.
If there is more than one of them, they may be expecting to do two rooms (or more) at the same time. For example, they may send a team of four of them to get the whole job done in a half day. Please phone and check.
When we had all the bedrooms done at once we cleared most things out but there was nowhere to put the bulky things - beds wardrobes and chest of drawers. We just emptied the wardrobes and drawers so they were as light as possible and the fitters shifted them from one side of the room to the other as the laid the carpet down. They didn't charge any extra.