Probably not that much. There are no primary sources that connect Byzantine Iconoclasm with Islamic Iconoclasm. Plus, the two Iconoclasms were quite different. The Byzantine version prohibited depictions of saints, while the Islamic version prohibited depictions of ALL living things. It seems that the connection between Byzantine Iconoclasm and Islam was pushed forward by the iconodules in order to portray the iconoclasts as non-Christians and heretics. The iconodule authors also connected Byzantine iconoclasm to both Judaism and Monophysitism. So, we can conclude that, due to big differences between them, Byzantine Iconoclasm was not modeled on the Islamic model.
Regardin the second part of OP - Σαρακηνoφρων / saracen minded - was a fairly common post iconoclastic slur used by iconophile ERE historians against those pesky iconoclasts. Leo IIi was called such, he started the first ERE iconoclastic period, nearly at the time of the iconoclastic edict of the Yazid II.