Fuel dyes are dyes added to fuels, as in some countries it is required by law to dye a low-tax fuel to deter its use in applications intended for higher-taxed ones. Untaxed fuels are referred to as "dyed", while taxed ones are called "clear" or "white".
Aviation gasoline is dyed, both for tax reasons (avgas is typically taxed to support aviation infrastructure) as well as safety (due to the consequences of fuelling an aircraft with the wrong kind of fuel).
Heating oil is any petroleum product or other oil used for heating; it is a fuel oil. Most commonly, it refers to low viscosity grades of fuel oil used for furnaces or boilers for home heating and in other buildings. Home heating oil is often abbreviated as HHO.
Most heating oil products are chemically very similar to diesel fuel used as motor fuel; motor fuel is typically subject to higher fuel taxes. Many countries add fuel dyes to heating oil, allowing law enforcement to check if a driver is evading fuel taxes. Since 2002, Solvent Yellow 124 has been added as a "Euromarker" in the European Union; untaxed diesel is known as "red diesel" in the United Kingdom.