Will LR backups work if I put the folder into the cloud on Dropbox?

Removed, posted to this thread in error.

In principle yes, but there may be a problem. Moving the catalog backup folder to Dropbox by itself does not create more room. The Dropbox folder is a local folder. You can however set the sync of any folder inside your Dropbox folder to 'Online only'. That will upload the contents to Dropbox and then delete the local copy (and leave just a kind of link). That is probably what you want to do. This should work for catalog backups too, but there may be a problem if Dropbox is too eager: Lightroom first creates the backup copies (there are two files now that are backed up), and then it zips them. If one of those copies has already been uploaded and deleted before Lightroom could create the zip, you may get an error. So try it to see if it works.

Despite the concerns from @Nikon ShootR™ , you don't actively try to access backup copies of Lightroom Classic, other than perhaps once every few years when there is a problem with your working catalog.

So, placing backups in DropBox seems reasonable to me (unlike placing the working catalog in DropBox, which is indeed a recipe for disaster).

 Along time ago when Lightroom was first a standalone product, and dropbox was just starting up, I used exacty the scenerio yoou are asking about.  It was a bed experience, as had serious difficulties keeping synchronized between the files I was editing within Lightroom and the new DropBox. I finally stopped using Dropbox for my production files.  Although I haven't tried what you're asking about in a long time, I had some terrible messes that caused me waste of time and some loss of media files.  I now only use DropBox for exporting and saving more "permanent" files.  
It is possble synchronizing is better now, however now that Dropbox has selective synchronizing, it all scares me a little.   I'd like to hear from others.