What are the circumstances that make Jihad obligatory on a Muslim citizen? What is the punishment for not taking part in Jihad when it has become obligatory?
As far as an individual is concerned, Jihad becomes compulsory upon him if both the following conditions are fulfilled:
i. The rulers give a call to each and every citizen of their country to participate in Jihad.16
ii. The individual himself is convinced that his rulers are waging war on the only legitimate basis on which war can now be waged: to uproot injustice and oppression perpetrated by a country. If a person is convinced that his country has some other objective in mind, he can of course refuse their call.
Regarding the second part of your query:
A study of the Qur'an shows that there is no worldly punishment for either a state or an individual if they do not undertake Jihad when it becomes obligatory upon them. The reason for this is that the Qur'an does not mention any such punishment. It seems that it is the Almighty Himself Who directly punishes such negligence in this world or in the Hereafter.
16. According to the Qur'an (9:38-9), in the Prophet's times, such a call was given by the Islamic state to the each and every Muslim before the battle of Tabūk.