It is possible that the solid aluminum hydroxide will not all dissolve when additional sulfuric acid is added during the third reaction of the alum crystal synthesis, in which case you should heat the solution. Why is this done?

Aluminium Hydroxide is amphoteric (compound that can react with both acid and base) in nature, hence it is possible that on addition of H2SO4, it will not form the white precipitate of Al2 (SO4) 3. In order to obtain the desired reaction, we need to heat the solution, which will lead in the disintegration of compounds into ions at rapid rate, leading to the formation of Aluminium Sulphate or Al2 (SO4) 3.