Valve seemingly have little interest in making a new Day of Defeat but they are at least letting someone else have a crack. Now on Steam Early Access is Day of Infamy [official site], a World War 2 multiplayer FPS heavily inspired by Day of Defeat - and made with Valve's blessings and support. Infamy is the work of New World Interactive, the folks who made modern military FPS Insurgency. They're trying to pick up where Day of Defeat left off, basically. Here, have a gander:
Like DoD, Day of Infamy is a class-based first-person shooter about fairly close-range infantry combat across the fronts of World War 2. It's not a serious sim but it's less arcade-y than many. Along with several 32-player multiplayer modes about capturing and holding points in different ways, it has 8-player co-op missions about attacking, defending, and patrolling territory, and a touch of singleplayer too.
Infamy started as a free mod for Insurgency but it's gone commercial with Valve's support. NWI are even making some Infamy maps based on Day of Defeat maps.
Day of Infamy is on Steam Early Access for Windows and Mac at £14.24/17,09€/$18.99, including a wee launch discount. NWI plan to release the full version some time from October 2016 to March 2017. Until then, then plan to polish and expand it with new maps, weapons, factions, and modes, and especially to improve the singleplayer side. You can follow their development roadmap on Trello.
If you want to see a little more, here's some alpha gameplay from a few weeks ago: