Firstly the reason for non-payment will be considered, for example a Dhimmi is excused from paying Jizyah if he is unable to pay it because of poverty. The below only applies on refusal to pay without any valid reason:
According to the majority of the mahabs: If a Dhimmi refuses to pay Jizyah then his treaty with the Muslims is broken and he reverts to being a Harbi. A Harbi can be killed, enslaved or ransomed at the discretion of the Muslims.
According to the Malikis, in addition to the above, when an adult male breaks his treaty of Dhimma, then the treaty is also automatically revoked from his women and minor children. Hence any women and minor children of the said person will be enslaved as spoils of war. The majority however do not void the protection from his women and children.
According the Hanafis the treaty of a Dhimmi refusing Jizyah will not be broken rather Jizyah will be taken forcefully from them. Note that the Hanafis differentiate between the initial refusal to accept Jizyah (at the time of the conquest) and refusal after initial acceptance.
The reason for all the above is that Islam commands waging war against the disbelievers until they agree to pay Jizyah. And war entails things like killing, enslavement, ransom, seizure of property and family as spoils of war etc.
قاتلوا ... حتى يعطوا الجزية عن يد وهم صاغرون
Fight ... until they pay jizyah with their own hands while they are subdued.
— Quran 9:29
فأمرنا نبينا رسول ربنا صلى الله عليه وسلم أن نقاتلكم حتى تعبدوا الله وحده أو تؤدوا الجزية
Our Prophet, the Messenger of our Lord, has ordered us to fight you till you worship Allah Alone or give Jizyah (i.e. tribute)
War was only stopped with the person because of his treaty in which he agreed to pay Jizyah and to be subservient to Islamic laws. If he now refuses to pay it he breaks the treaty and hence the war resumes, his life and property loses sanctity and he is treated like a prisoner of war.
Ref: Mawsoo’ah al-Fiqhiyyah 7/138, 24/157 and Fiqh al-Islami wa Adillatuhu 8/5885