TODO: Could use more info from Doom TASers.
Doom is unique in that it is one of the first, if not the first, games to have their own input movie playback format in the form of demos. This lead to it being the first game to have tool-assisted speedruns as its save files effectively act like save states. With the release of the Doom source code, source ports have been created to facilitate the creation of speedruns, both real-time and tool-assisted.
A demo should at the very least be able to play back on the source port it was made on without any problems. A lot of liberal source ports such as ZDoom are not concerned with demo compatibility, often to the point where demos aren't even guaranteed to play back correctly on the same version it was made on. In fact, some don't even support demo creation. Thankfully, there are a number of more conservative source ports that that do guarantee correct demo creation and playback.
Taking all of that into account, demos made for WADs that are only supported on source ports that don't have robust demo playback are not accepted.
Source Ports for Tool-Assisted Speedrunning
This enhanced but conservative source port has a special feature that allows it to play back demos made on all released versions of vanilla Doom, along with other source ports it was derived from. It also has a strict compatibility feature that allows any demo made under a specified compatibility level to be played back on more than just PrBoom+ itself. (e.g. A compatibility level of "Doom v1.9" allows the demo to be played back on vanilla Doom 1.9.) In addition, it has built-in video dumping capabilities, making it the only approved option for encoding Doom demos.
This source port is a fork of PrBoom+ that's geared towards speedrunning. In addition to the slowdown feature PrBoom+ provides, it also has a rewind feature, on-the-fly map restarting, and parameters that provide statistics to aid in speedrunning, both RTA and TAS.
This is not actually a proper source port, but a demo editor that's like BizHawk's TAStudio. For tool-assisted speedrunning, it is used for very precise changes to input down to the tic.