It seems most Protestants support the State of Israel and/or the Zionist cause. What theologically distinguishes those who do not?
Under Covenant Theology, the members of the modern-day church are viewed as the true children of Israel. The promises made to Abraham regarding his descendants, therefore, apply to the church.
Under Dispensationalism, the modern-day Jews (i.e. the law-practicing blood-descendants of Abraham) are still considered to be the children of Israel. The promises made to Abraham regarding his descendants, therefore, apply to the Jews, and consequently, the Jews living in the modern-day country of Israel.
One of those promises which is commonly brought up is:
And I will bless those who bless you,
And the one who curses you I will curse.
Based on that promise, support for the State of Israel, to the dispensationalist, is a religious matter. It is not uncommon for them to believe that if they support the state of Israel, God will bless them, and if they do not, God will curse them.
For an adherent of Covenant Theology, however, it typically becomes a purely political judgement of foreign policy. There is no religious overtone to their decision as there is with the dispensationalist.
So, with that background in mind, to answer your question more specifically, Christians who strictly adhere to Covenant Theology have no religious motivation for supporting Israel. If such people do support Israel, it is not due to their Christian doctrine.