The sun is a star that, like every star, emits its own light. The twinkling of stars is due to the atmospheric refraction of light, making it a point source of light. The starlight gets refracted dye from the difference in air density at various atmospheric levels when it enters the earth's atmosphere. When the refracted starlight comes towards us, it appears brighter compared to when it comes less towards us, and the star appears to be twinkling at night. 

Planets appear closer to earth and larger in size than stars, resulting in them not twinkling. These are point-size light source collections. The average of the dimmer and brighter effects produced by the different parts of planets is zero, thus the twinkling effects get nullified and they don't twinkle.