The secret to white vinegar's cleaning ability lies in its acidity. The acid helps eliminate sticky substances such as carpet adhesive but vinegar isn't acidic enough for the fumes to be toxic or for it to burn the skin. Spray the carpet adhesive with pure white vinegar and let it sit for at least a minute or two. Scrub at the adhesive with a damp scrubbing sponge and the adhesive should come off without damaging the tile. Use a soft sponge and not an abrasive material such as steel wool to prevent tile damage during the cleaning process.
Liquid dish soap is a natural degreaser that works as a gentle, non-toxic cleaner on numerous surfaces, including tile. Mix several drops of dish soap into a gallon of warm water. The hotter the water, the more effective it will be at removing the adhesive from the tile, but avoid using water so hot that it burns you. Dip a sponge into the solution and then scrub at the adhesive. Once it is fully removed, rinse the tiles in regular water to remove any remaining traces of soap.
Rubbing alcohol is a very effective cleaner and though it is harmful if swallowed, it cleans efficiently in small amounts and evaporates quickly so it doesn't pose a health risk to kids or pets who are around while you're using it. Hold a sponge up to the bottle and then turn the bottle slightly over so the alcohol starts to permeate the sponge without actually spilling out. It should then scrub away the carpet adhesive with little difficulty. Rubbing alcohol can make tiles slippery, so rinse them once you're finished to prevent any slipping risk and then dry them with a towel.
Sprinkle baking soda over any carpet adhesive that remains. It helps make the adhesive less sticky and makes it easier to remove. It also adds friction when you scrub at the adhesive with the scrubbing sponge, which is an added benefit. The sponge should be damp with warm-to-hot water for best results since a dry sponge won't work well with the baking soda and may scratch the floor. The baking soda fizzes when it comes into contact with water and this also helps loosen the adhesive. Rinse the floor with regular water once you finish.