The things that appear in the red warnings are usually things like odd or impossible dates in the database fields at the top of the screen, the creation of new categories, or broken tags in your text - technical errors, in other words. It's not a judgement of the content of your biography per se and doesn't evaluate the quality of your work.
Ideally, of course, you should pay attention to the message in the warning and fix what the warning is telling you to fix. If there's no "Save anyway" button, then you won't be able to save your work until the error has been remedied.
You can, if you get a bold red warning with a button labelled "Save anyway," still save your work (but maybe first make a comment in the biography that it's work in progress; then people will know that you're not done yet). If you don't click that, then the changes you've made won't save.
If you're in a situation where you've done a lot of work on the biography, and you get a warning or error that you don't know how to handle, then as a safety-measure, I would click into the edit window and copy all the contents out of it and paste it into a different piece of software, e.g. Notepad or similar, and save it to my computer. That way, in case something does go wrong, you still have a backup of your work which you can paste straight back into the edit window later.
