Adding any gain is destructive in the sense that you raise the noise floor in your recording. That's why you should aim to get the levels 'hot' enough so that you limit any reason to do anything in post. [/preaching to choir]
I have trouble with that explanation because cranking up the gain via pre-amp also raises the noise floor.
For the purposes of discussing this issue, let's focus on 24 bit. Without a doubt, there is a loss of resolution when 16 bit recordings are made below peak, etc.
First of all, "destructive" does not mean "degrades the sound." It means the original file is forever altered, be it for better or worse. Non-destructive editing means you get a new file and the old one is left alone (or is completely restorable). If you process a gain change on the original file, it's destructive. If you generate a new copy with higher gain, the gain change was non destructive. This is a red herring.
Freelunch is right. You don't add any extra noise by doing a gain change. The signal to noise ratio stays the same. If you normalize and then turn down the playback volume so you're listening at the same levels as before you normalized, you would have the same amount of noise as if you didn't normalize. That's equally true of 16 and 24 bit, although the 24 bit
may have a lower noise floor, so the bit rate is not a data point. (There's actually more to this, but enough for now.)
So why normalize?
Because if the levels are lower you have to turn up the playback stereo further to get the desired volume. Then, not only are you turning up the noise on the recording just as much as you would if you had normalized it, but you are also turning up the noise generated by your playback system, adding to the total noise.
Plus, it's just annoying when one CD plays at a lower level than others and it sounds like a less professional production.
The above refers only to peak normalization, which is what normalization generally, but apparently not always, means. Normalization with compression has a whole set of different issues.