Q. Two situations one question: Are you allowed to erase Shem HaShem on a cellphone?
1 – am I allowed to use a picture of a brocha on my phone that has Shem HaShem on it?
2 – what to do if, for example, someone has posted a picture of something with Shem HaShem in it.

A. Most Poiskim rule that Hashem’s name written on a screen can be erased and they have no sanctity as the pixels which make up the letters on a computer screen are being refreshed many times a second and are considered to be more virtual than real.

T’shuvos Vehanhogos (3: 326) permits erasing Hashems name adding also the fact that it was not written with the intention of kedusha. He also points out that s’forim like chumashim and sidurim, are commonly printed today by using computers and so are the very effective and essential programs to check the kashrus of sifrei Torah and tefilin. So too is lenient Avnei Yoshfe (4: 105) and Nishmas Shabbos (Mechika Uksiva 137).and others (see Piskey Teshuvos 174)

However some Poiskim are more stringent, Shevet Halevy (7, 4) rules that even though erasing a name written on a screen is not biblically prohibited one should avoid doing so only in cases of great need, a similar opinion is to be found in Sheilas Shoul (Y.D. 59: 3).

Horav Shlomo Miller’s Shlit’a opinion is that it is permitted to read Torah and Tefilos with Hashem’s name from a screen when needed, but one should always do so with decorum and kedusha.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit”a