The advent of the information age, and the proliferation of computers as a medium of communication and study raises questions which our tradition could not have anticipated, yours among them. Nonetheless there is a body of discussion on the subject of destroying the Name of God that is helpful in approaching this most modern issue. An overview of the traditional issue from the Reform perspective is found in Current Reform Responsa, edited by the late Solomon Freehof, and published by HUC Press in 1969. Your local Rabbi or synagogue may have a copy.
I have not seen a more recent discussion, though the Computer Committee of the Central Conference of American Rabbis may be discussing the issue. The prohibition of destroying God's name stems from a passage in Dt. 12:3-4, where we are commanded to obliterate the names of idols, "but you shall not do so to the Name of the Lord your God." Through the centuries, the rabbis expanded the prohibition to include the Name of God (YHVH) in all sacred writings. But they created exceptions to the rule. Two of them, in my opinion, apply directly to your question.
First, the Mishnah (Shabbat 12:4) concludes that writing done with any material that does not endure is not actually writing. The Name of God created in such a way could thus be erased without violating the law. Since electronic images are not permanent, they are only digitized electronic impulses, and thus are not actually writing as the sages defined it, erasing the Name of God from a computer screen or disk would not violate halakhah. Second, the rabbis debated whether the prohibition of erasing God's Name applies when the Name appears in a text which has not been consecrated for a religious purpose. Citing the Shluchan Aruch and a number of later halakhic works, Rabbi Freehof concludes: "It is the opinion of many of the earlier and later authorities that no sin is committed by erasing a Name which we know for certain was not consciously consecrated.
Again, this all applies to the Shem Ham'forash, the Tetragrammaton Yod Hey Vov Hey. It seems to me that both of these principles apply to the Name of God "written" on the computer. Electronic images are temporary and not writing at all. as halakhah defines writing; and the Name of God in the computer, I presume, has not been consciously consecrated. Under these circumstances, erasure of the Name would not be a violation of Jewish law.