Why doesn't this image save as web?

Perhaps it would be useful to 'Save As' .gif or .jpg file first!

It's important to retain as PSD and not work from a gif or jpg.

As far as I know, .psd files cannot be displayed on a web page

You don't understand the issue.  Please re-read the OP.

As you have discovered, you can't get to a jpg or gif from a 32-bit image in Photoshop. It has to be down sampled to 16-bit first.

When you go to downsample, Photoshop opens up the HDR Toning dialog so that you can tell it what to do with all of those extra colors.

I tried it with your sample and yes, the default settings give disastrous results.

Try this instead. Make a duplicate copy of your file and open them both in Photoshop. Then arrange the windows horizontally so that you can see both images at the same time. Now, in one of the files, go to Image>Merge>16-Bits/channel.

Because your image is not on a background layer, Photoshop will ask you if you want to merge. Select merge.

Now the HDR Toning dialog will open. In the Method pull down box, select Exposure and Gamma. Now adjust the gamma slider until you get a close match. (you may have to tweak the exposure slider a bit as well)

Click OK. You can now use the "Save for Web and Devices" to save a copy of your image to your desired format.

Yes - I've been using the HDR Toning window to adjust the colors and that has worked well.

But I'm wondering why this image started off at 32bit?  It's a collage of other image files.  I guess I probably started off with a 32bit host image?  

Would 16bit cause the same issue?  If I want a new image to have the "save to web" option, which bit size should I start with?

That image was pre-existing so I'm not sure how it was created.  It just dropped into PS.