Scholars Privilege

The picture shows Jewish religious soldiers at prayer. These mainly come from the National Religious sector which is parallel to Modern Orthodoxy. They comprise less than 20% of the inductees yet contribute about 35% (or more) of the front-line volunteer units and an almost equal percentage of the NCOs and officers [for more figures see Reply by Yair Davidiy to Comment by Benjamin Hurwitz below]. Nevertheless their contribution is not felt to be appreciated. Their religious rights are not always respected. They also loose a not insignificant proportion of their soldiers to the forces of secularism. The Ultra-Orthodox for their part do not suffer from such losses. That is not the reason they are loathe to serve in the IDF but it does place it in a positive light when considered in line with other considerations discussed below.

The question above should perhaps be re-formulated as to something like:
Why do Jewish Orthodox males refuse to be drafted to serve in the Israel military?

As I understand it there are several different reasons that overlap and interlock with each other:

1. In Place of the Tribe of Levi. Jewish Ultra-Orthodox males to a large degree learn Torah full time. They are supported by their extended families and by contributions from the public, philanthropists, as well as, to a modest degree, by the government. They are fulfilling the task of the Tribe of Levi in ancient times. The Levites would serve the public in religious matters. Unlike the other tribes, Levites would not go out to war against the enemies of Israel.

Numbers ch. 26:2-51 gives a census of all the Tribes (apart from Levi) from 20 years up to 60. According to Scripture 20 years was the age an Israelite male began his obligation to do military service. The Tribe of Levi is not numbered in this census. The Tribe of Levi is however counted on its own (Numbers 26:52-62). The counting of Levi begins from a month old. Levi was not listed from the age of 20 because the Levites did not go out to war with the rest of Israel.

Maimonides (Laws of Shemitah and Yovel 13.12):

... they do not go to war, as do the rest of Israel, they do not receive an apportionment of land as an inheritance, and they do not obtain things for themselves by the strength of their bodies; rather, they are the Warriors of the Almighty, as it is said: 'Bless, O LORD, his Prowess,' and He, Blessed be He, provides for them, as it is said: 'I am your portion and your share' (Numbers 18:20).

That is the main reason. Due to the Holocaust and the pressures of Modern Life Religious Jews are fighting a war of survival physically and ideologically. They who learn Torah are in the front lines of this struggle. It is felt that learning Torah brings merit on Israel. The IDF up until now has benefited from Divine Intervention on its behalf. This may in part be due to Jews learning Torah.

Other reasons include:

2. There is an anti-Religious ideology that permeates sections of the IDF command. Military service to some degree in the past was turned into a tool to modernize and secularize Jewish youth. The Ultra-Orthodox naturally do not want to be subject to such influences.

3. Objective difficulties. 
 Military Life even without an anti-religious agenda can makes strict observance of the Law difficult.

Other points worth noting are:

4. The IDF does not really want Ultra-Orthodox (Hareidim). Leading officers have said as much. The peculiar requirements of Ultra-Orthodox soldiers may cause more complications than the effort is worth. Political pressures to force the conscription of Hareidim do not emanate from military sources.

5. The IDF has more soldiers than it needs. The IDF suffers from under-utilization of the manpower already under its command. Boys and girls are inducted into service more for ideological and social reasons than defense needs. The IDF might be better off relying on full-time professionals in all ranks and functions.