After Zenimax had acquired Id Software in 2009, preparations for an E3 demo of DOOM 4 was underway. While it is unknown if the demo was set to happen during E3 2010 or E3 2011, we do know some of what was featured in the demo thanks to information from anonymous staff members acquired by Crispies during the late 2010s, folder trees from a build that used to exist on Steam's databases and what footage has showed up in the years since then. What follows below is a summary of the demo as best as we can speculate with the current information.
The E3 demo was a vertical slice (development term for a level or build that aims to show a short summary of what the game will be like, used for pitching and/or for floor demos) containing the level "Ashes" set right at the start of DOOM 4's story. The player is a civilian in New York who witnesses a "Hellwave" erupt from a large structure, tearing the city up and transforming people into monsters. He begins making his way towards the headquarters of a resistance group that have been part of an uprising against military action during a quarantine. After being knocked on his ass by the Hellwave, the player roams the streets of New York and enters an apartment building where he's armed with a double-barreled super shotgun. As he leaves that building he sees a library split in two in what appears to be the big splash shot moment of the demo. He makes his way inside the library, fighting Former Human enemies along the way. Eventually he makes his way into the sewers, where he must fight a pair of Imps for the first time. They can teleport in addition to their classic fire ball throw, as well as climb walls and ceilings similar to their DOOM 3 appearances. Taking out the Imps, the player continues making his way to the subway tunnels of New York and eventually up through a station. From the subway station the player makes his way to a massive apartment structure that serves as the base for the resistance group. Likely serving as the end of the demo. In addition to the super shotgun, an assault rifle and a chainsaw has also been seen in shots of the "Ashes" level.
Here are screenshots from various sources presented in a specific order to give you an idea of what the demo might have looked like. Most of the screenshots are from level designer Thao Le's portfolio and as such don't depict gameplay but rather just shots of the level without any actors in it. We don't know where enemies and such would appear outside of the few seconds of footage that exists of the level.