What is the ruling on leaving used newspapers, which may contain the name of Allah, thrown in the street or used by others?

Glorifying the name of Allah is part of glorifying His rituals, which indicates piety of hearts. Part of glorifying His name is to protect it from being thrown in the street or used in a derogatory manner.

However, this should be done within the limits of the Sharia objectives, which include removing hardship and difficulty from people.

What you mentioned is a widespread problem in this era, and it is difficult to guard against it. Therefore, you should do what does not cause you hardship concerning these newspapers. If you know that they contain something revered in Sharia, such as the name of Allah, the Quran, and so on, then you must protect them. You are not required to search every piece of paper you encounter, as the difficulty in that is obvious.