It wouldn't & didn't.
The Silence had deliberately set out to create a "fixed point in time". They succeeded in doing so but the event that constituted the "fixed point" wasn't the event they intended & believed it to be. Instead of River shooting the Doctor being a "fixed point", River shooting the Teselecta was the "fixed point".
What caused the entire timestream to collapse in "The Wedding of River Song" was her failure to shoot the Teselecta because she thought it was the real Doctor. She tried to rewrite a "fixed point in time", which cannot be done. Once the Doctor had shown her that he was safe inside the Teselecta (by letting her see him in its "eye"), she went ahead & shot the thing, thus restoring the timeline.
It was because it was the Teselecta, not the Doctor, that Canton & River (in "The Impossible Astronaut") made sure the "body" was burned before it could be examined closely. As the Doctor later explained, the Teselecta wasn't harmed. It sank to the bottom of the lake & (once nobody was watching) simply walked out, with all aboard -- including the Doctor -- safe & sound.