Those who were jealous were characterized by feelings of insecurity and being envious of attention that was given to another.
The Eighth Doctor once theorized that the Master's hatred of him stemmed from his old friend's jealousy at the praise that the Doctor had received at the Time Lord Academy in their youth. (AUDIO: Masterplan)
Rose Tyler had a notable jealous streak when it came to the Doctor, Jack Harkness noting this when Tara Mishra joined the TARDIS and bonded with the Ninth Doctor. (COMIC: Slaver's Song) Her jealousy was far more pronounced when she travelled with the Tenth Doctor, being very offended to learn that Sarah Jane Smith, and other females, had travelled with the Doctor before her, something Mickey Smith called her out on. (TV: School Reunion) Such was Rose prone to jealousy that she was targeted by the Iagnon, which fed on the emotion. To save her life, the Doctor deliberately exacerbated her jealousy, overfeeding the parasite. (COMIC: The Green-Eyed Monster)
Following Rose's departure, the Doctor's overt mourning of her caused his next companion, Martha Jones, to become jealous of the absent Rose. Even Jack Harkness, rather bitter at the Doctor abandoning him on Satellite Five, jealously opined that the Doctor hadn't abandoned Rose on account of her being blonde. (TV: Utopia) Ironically, Rose would become jealous of Martha during the 21st century Dalek invasion, feeling left out when the Doctor's various companions were communicating via the Sub-Wave Network. (TV: The Stolen Earth)
Missy expressed jealousy when the Twelfth Doctor named Davros as his archenemy, swearing to "scratch [Davros'] eye out", (TV: The Magician's Apprentice) jabbing him directly in it when the two finally met. (TV: The Witch's Familiar)
The Doctor themselves was also prone to jealousy, the Eleventh Doctor becoming visibly jealous at Porridge expressing attraction to Clara Oswald. (TV: Nightmare in Silver) When the Twelfth Doctor took Clara to meet Robin Hood, he was transparently jealous of the attention she gave the archer, the Doctor constantly trying to outdo him, much to Clara's exasperation. (TV: Robot of Sherwood) He had a similar reaction when he later met River Song's other husbands, King Hydroflax and Ramone, and that she'd had more spouses across her life. (TV: The Husbands of River Song)
Historians who studied the Dalek race developed a theory that the Thousand Year War on Skaro was instigated by the Kaleds feeling jealous of the Thal Dome, building the Kaled Dome in response and leading to a cold war that eventually turned hot. (PROSE: Dalek: The Astounding Untold History of the Greatest Enemies of the Universe)