How do I remove old glue/linoleum backing from wood floors?

Try goo-gone

Hi Vickie, hopefully this article will help you get the old glue off - https://spotremoval.coit.com/how-to-remove-glue-from-wood

Have you tried heating up the area with a hairdryer and tried carefully scraping? (Or a steam iron held very close to the backing without touching, then scrape?)

The method to remove glue that I would use, would be to get a heat gun and scrape away any of the old glue and backing after heating. After you have gotten as much of that scraped away, let it dry again and then sand with fine sandpaper to remove the rest of what was left behind.