Let's say that the ISS has to be immediately evacuated and that one of the two Soyuz is broken.
Could the crew quickly remove "unnecessary weight" (seats maybe?) and fit six person to return back to Earth?
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No. And it doesn't even matter that there's no space for a crew of six no matter how small they might be or if they threw away some other pieces and try to make do. The life support of the descent module of the Soyuz TMA can only support up to three people. Since descent time is more or less constant regardless of the descent mode, and there's no way to descent faster after module separation without killing the crew in the process, you could only transport additional crew of three if they don't depend on module's available life support's provisions. I.e. if they're already dead and essentially cargo.
They wouldn't fit anyway, Soyuz TMA descent module can fit 50 kg of cargo for a crew of three or 150 kg for a crew of two, with total capacity of 305 kg. And with habitable volume of only 3.5 cubic meters, it gets pretty tight in there with all the essential equipment, spare chute and astronauts / cosmonauts in their Sokol IVA suits as it is:
ISS crew members, U.S. astronaut Daniel Burbank and Russian cosmonauts Anton Shkaplerov and Anatoly Ivanishin,
crowded inside their Soyuz capsule shortly after landing in Kazakhstan, on April 27, 2012. (Reuters/Sergei Remezov)
I found an actual reference for this, although it's in a memoir, not a technical book. Jerry Linenger's book Off The Planet states
Unfortunately there is no way to squeeze more than three people into a Soyuz capsule. There are only three seats in the spacecraft. Even if one other person, without a seat, sprawled across the sitting crew, the presence of that fourth person would shift the center of gravity of the vehicle, most likely causing the capsule to tumble during reentry. A tumbling vehicle means that the capsule's heat shield would not be pointed in the proper direction.
Page 112 of the McGraw-Hill 2000 printing.
This was in reference to the fire emergency on Mir during the Mir 22-23 crew handover period. There were two Soyuzes docked and six people on board. The fire blocked access to one of the Soyuzes. Linenger concludes that it would have been impossible for all six to evacuate in the accessible Soyuz.
According to the NASA fact sheet on Soyuz landing, it takes about 3.5 hours from undocking to a landing on Earth. But in the three first hours they have the otbital module, and you can likely then cram up to six people into Soyuz.
The Soyuz has a cargo capacity of 50 kg in addition to three crew members. If you dropped the Sokol suits, which are only a backup system, you can fit in one more without going over limit. But how the mass is distributed is also important, as the capsule has to be balanced
As for volume, it is pretty tight though. Is it enough space to fit one more person in the descent module? Finding pictures which clearly shows the spare room is difficult, so I decided to do some original research. As it turns out, crowding a 3.5 cube meter room with four people is not all that difficult. The spare room of the capsule is also pretty much concentrated between the commander and the hatch.
Surviving in a closed space is always a question of CO2 removal. Assuming worst case, that the system can only support three people, carbon dioxide from the forth crewman will build up in the atmosphere of the capsule. At normal human carbon dioxide emission, the concentration after half an hour would be between 0.5 and 1 %. That is still within acceptable limits. Using the equation found here, I get a comparable ressult.
The Soyuz has custom fit seats for the Cosmonaut, and that indicates high acceleration is a big deal. But like @SF said, they are likely to get away with it with only broken bones or minor damages.
In conclusion, it is not possible to land the Soyuz safely with a six person crew, but it may be possible to fit in one more.
This is not to say that the Soyuz can actually carry four people, but it may be possible in theory. Real world safety procedures will never allow this as it has a too high risk.
Even emergency events have never been enough for trying to stretch the capacity of the spacecraft.