All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.
The scholars differed in opinion about the Arabic letters and whether or not they are respected for themselves. We favor the opinion that they do not have any sacredness especially for the necessity in our time and age.
However, this difference of opinion should be considered and avoided as much as possible, especially if what is written is a respected name, such as the names of the scholars and the names of sacred places like Makkah and Madeenah.
The view that we adopt here at Islamweb is that throwing away what contains such names is makrooh (disliked) for tanzeeh (i.e. close to halaal [permissibility], unlike makrooh for tahreem which is close to haraam [prohibition]).
Therefore, you should – but you are not obliged – get rid of things on which the name of Makkah is written either by burning them or by wiping out the Arabic writing or a similar method, and you should not throw them in the trash though you are not sinful if you did do so.
As regards what is not considered disrespect, such as putting it on the floor in a clean place and the like, then there is no harm in doing so. Likewise is entering the toilet with such things, as what is disliked is the things on which there is mention of Allah.
Bear in mind that hardship entails facilitation and that Allah does not burden a soul beyond its capacity.
Allah knows best.