All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad  is His slave and Messenger.
  is His slave and Messenger.
It is not permissible for a non-Muslim to be a member of the Board of Directors of an Islamic school because this is a kind of giving them positions of power and authority (over Muslims). Ibn Taymiyyah  after mentioning the verses which prohibit from taking the non-Muslims as allies/friends, he said: "It is for this reason that the scholars of the Salaf (righteous predecessors) used these verses as evidence for not seeking their help and assistance in the matter of Wilaayat (i.e. in positions of power and authority)."
  after mentioning the verses which prohibit from taking the non-Muslims as allies/friends, he said: "It is for this reason that the scholars of the Salaf (righteous predecessors) used these verses as evidence for not seeking their help and assistance in the matter of Wilaayat (i.e. in positions of power and authority)."
Moreover, Imaam Ahmad  reported with a sound chain of narration from Abu Moosa Al-Ash’ari
  reported with a sound chain of narration from Abu Moosa Al-Ash’ari  who said: "I said to ‘Umar
  who said: "I said to ‘Umar  : I have a Christian clerk." Then ‘Umar
: I have a Christian clerk." Then ‘Umar  told him: ‘‘What’s wrong with you, may Allaah destroy you! Did you not hear that Allaah Says (what means): {O you who believe! Take not the Jews and the Christians as friends or allies.}[Quran 5:51] Why did you not take a Muslim to do so? Abu Moosa
  told him: ‘‘What’s wrong with you, may Allaah destroy you! Did you not hear that Allaah Says (what means): {O you who believe! Take not the Jews and the Christians as friends or allies.}[Quran 5:51] Why did you not take a Muslim to do so? Abu Moosa  replied: "O Ameer-ul-M'umineen (commander of the believers), I benefit from his writings and I do not have anything to do with his religion." Thereupon ‘Umar
  replied: "O Ameer-ul-M'umineen (commander of the believers), I benefit from his writings and I do not have anything to do with his religion." Thereupon ‘Umar  said: "I would not honour them while Allaah humiliated them, I would not value them greatly whereas Allaah debased them, and I would not bring them closer to me while Allaah distended them (from Him and from His Mercy)."
  said: "I would not honour them while Allaah humiliated them, I would not value them greatly whereas Allaah debased them, and I would not bring them closer to me while Allaah distended them (from Him and from His Mercy)."
Furthermore, when Imaam Ahmad  was asked about hiring a Jew or a Christian to work for Muslims, such as in collecting land tax, he said: "We do not seek their help in anything".
  was asked about hiring a Jew or a Christian to work for Muslims, such as in collecting land tax, he said: "We do not seek their help in anything". 
Allaah Knows best.