Is it possible to save a File Output node in the compositor without rerendering the entire scene?
I don't think there is a way to make the file output node save the files it is configured to save without rendering the scene.
A possible workaround for limited cases is to reroute the image input to a viewer node so that the image data can be viewed in the UV/Image editor. This allows the image data to be manually saved to a file.
Obviously for one frame of a simple multiple image file setup this is fairly easy, if you have configured a multilayerEXR then this will only allow saving one changed layer, one frame at a time. You could then open this file and replace the specific layer contents of the intended target file. The compositor could be used to combine the adjusted layer. Or you could use an empty scene that opens the existing files and composites the alterations into new files.
Since you don't want to rerender, I am guessing that your scene is already rendered, but complex. I think I may have an alternate solution for you.
I don't think that you can do what you want, but you can open a UV Image Editor and select Render Result from the list of images in the bottom panel.
Paul Gonet's image on BSE here.
From here, you can easily save.
You can substitute a Viewer node in basically the same setup as previously described. In the UV Image Editor, select Viewer Node from the menu.
Also make sure that you have Backdrop checked at the bottom of the Node Editor.
From here, you should be able to connect any node to the Viewer node and have it show up in the UV Image Editor.
I would have thought 'do not overwrite' would achieve this, but nop. [Or if you turned off the camera icon for 'render' but of course that is it's visibility in the render.]
The only solution I ever found was to have a second file that loads the images files from an Input Image node. Silly really. Scenes are the only way otherwise.
It seemed hinted here that you can force the Viewer node to "Save this Image" in the node inspectors Item panel, which might qualify, but again I just make a separate file for compositing typically.
It seems you are also able to duplicate Scenes and change the settings of things there. I think you can make an empty scene and then the 'render layers' input node in the compositor can be used to select different scenes and render them with a little button on the node.
you can use turbo tools to do this: