Today we're using De-Solv-It Contractor's Solvent to remove the sticky, left over from removing the vinyl floor tile. The floor tiles leave behind one heck of a sticky mess, leaving one with very few options. Option 1, use De-Solv-It Contractor's Solvent (or other similar product), let the product work itself into the sticky stuff, then scrape it all away. Repeat the process as needed, mopping between attempts does help dramatically.
Option 2, sand sand sand. You'll probably gum up a lot of paper if you try to sand all that sticky off. It'll work though, just going to cost you more time, more headache and more sheets of sandpaper. Option 3, rent a drum sander. Rental cost is about same as the Solvent itself. You'll have to buy the sand paper for it as well. 20-30 minutes for a small room is enough to rough off all the sticky and leave the floor mostly sanded. Leaving you with Option 1 and/or Option 2 for the edges and corners (which will seem way worse than they are, because of how easy the rest of the floor was).