It's not quite Left 4 Dead 3, not yet, but a new title in the series is indeed on the way. Left 4 Dead: Survivors is a game being designed by Valve and Taito exclusively for Japanese arcades.
This isn't the first time that Taito has taken one of Valve's properties and turned it into an arcade game for Japan- they did the same with Half-Life 2: Survivor, which was released in 2006. The arcade cabinet features a chair and what seems to be pretty typical first person controls, simplifying the game and adding a timer in order to make sure they get more money out of you.
Left 4 Dead: Survivors seems like it will follow in that title's footsteps, although there's no real news available for it just yet, not even any confirmation that the game will feature its trademark four-player co-op gameplay. But the above image was just spotted on Nesica.net and a live action video of the game that Taito released back in February is embedded below, from back when the game was known as Project Z.
As for Left 4 Dead 3, it's still rumored to be in the works. Valve certainly isn't going to be letting go of one of their finest series just like that, although they've been as tight-lipped about it as they usually are about their upcoming titles (Half-Life 3, anyone?) For now, Turtle Rock Studios's Evolve looks like it will fit the bill for anyone looking for a similar type of game, with a hugely improved multiplayer versus mode that pits four hunters facing one player controlling a gigantic monster.