The Warp Chair is the alpha version of the Messenger Lamps. It functions almost the same with slight differences.
There is a phantom sleeping in the chair. In order to use the Warp Chair, the hunter must "Shake the phantom Awake”, similar to igniting the lamps in the final game.
The Hunter will sit on the chair and sleep to be transported back to the Hunter's Dream.
Even in the fully released version of the game, Micolash and other Mensis scholars sat on their Warp Chairs to be transported into the nightmare and to have an audience with Mergo.
This mechanic may have served as an allegory for the constant themes of awakening, perhaps to reinforce that both realities in the Hunter's Dream and Yharnam are in fact dream realms that the players visit. The idea of dreams within dreams is still present in the nightmare realms and Chalice Dungeons.