Trouble removing vinyl tile and underlayment from wood flooring
Sometimes it takes a village. Here's a summary of what ended up working:


The first layer of tiling came off without too much trouble by using a heavy duty paint scraper (with a slight bend to it) and a hammer. I was able to get through this in less than an hour.
The second layer was much harder. I believe @Ecnerwal was right in that it was actually a layer of older vinyl or linoleum. This ended up coming up only with the use of a heat gun and the scraper. This took 3+ hours.
The remaining glue (putty like material) responded well to hot water. I heated up a kettle and would put a small amount of hot water on the remaining material. I used the scraper and a piece of steel wool to remove the material, sometimes requiring two passes. I had a towel handing to wipe up the water and material.


I'm now looking at fir flooring in relatively good condition, minus a nasty gouge mark I made when first trying to remove the tiling by force.

Some other details for those reading:


Make sure to take precautions as there may be asbestos in your flooring. Do not assume any of my steps taken were safe.
Paint thinner, as recommended by @Ecnerwal did not have any effect
I tried using some orange-based product to remove the glue from the floor. It had a minimal effect, took longer, and was more expensive than hot water. I didn't let it sit longer than a few hours, which may have limited it effectiveness (label recommended 30 minutes to 24 hours).


This, in the end, was really backbreaking work. Whether it was worth it or not will be more obvious once I figure out how well I can repair the gouge I made.

Thanks everyone for your help with this!