Cut into the linoleum with a utility knife, and peel up as much of the tile as possible. It will likely come up in pieces rather than in whole tiles.
Place old towels on top of the mastic and linoleum pieces. Pour boiling water onto the towel, and allow it to sit for several minutes to soften the glue.
Work a razor scraper under the linoleum, and scrape up as many of the tile pieces as possible. Reapply the towels and boiling water as needed.
Use dry heat to soften the glue if the boiling water method proves unsuccessful. Aim a heat gun or hair dryer on a high setting at the mastic. Work the razor scraper under the linoleum pieces and scrape them off as the glue softens.
Apply a chemical-based solvent to stubborn spots. Wear gloves, ventilate the area well and use the solvent according to the manufacturer's directions. Work in small areas to avoid excessive exposure to the chemicals.