Are you intending to refinsh the fir? Sanding will take the layer off.
If the fir is finished, heat should work by loosening the finish underneath. The finish and the cardboard will come up.

We are intending to refinish the floor. The cardboard/adhesive layer is pretty thick. What kind of sander do you recommend? If it's a powerful hand sander, would we run the risk of creating an uneven floor?

The fir is not finished.If we tried heat, what form? Thanks!

I got some scary gunk off of my fir subfloor by using the Silent Paint Remover, it's an infrared heater. You hold it over the surface for a number of seconds, and it softens & bubbles up the layer enough to get a nice sharp scraper underneath it to scrape it up. Tedious, but it worked very well without damaging the floor or creating any low spots or dips. I sanded the fir for refinishing afterwards.

The SPR is very pricy at $390, but I was able to resell mine almost at cost -- look on eBay etc for a used one perhaps? And in all honesty, if you have an infrared space heater, I bet it would work just as well. Just exercise caution, as always!

Here is a link that might be useful: Silent Paint Remover

Before you sand, have you had the adhesive under the cardboard layer tested for asbestos?

Before you break out all the harsh chemicals, you might try some hot water and a little vinegar. You don't have to pour water all of the place - just spray or mop it on and wait for the cardboard to absorb it. Do a little test area and see if it works.