What is the ruling on drinking from a cup that has the name of Allah written on it, and how can it be disposed of without disrespecting the name of Allah?

We do not know of any evidence that prohibits drinking from a cup inscribed with the name of Allah. The basic principle regarding things is permissibility. This is not considered a desecration of Allah's name, just as the Prophet, peace and blessings of Allah be upon him, used a silver ring engraved with "Muhammad, Messenger of Allah." If the name of Allah is removed by erasure or scratching, that would be preferable. It is obligatory to protect the cup from desecration, such as placing it in an inappropriate location or taking it into the restroom. If it becomes damaged, what contains the name of Allah must be preserved by burying it in a pure place or by destroying it through burning or breaking.