What is the ruling on someone who, as a child, threw coins bearing the name of Allah in a place full of papers, mud, and plants? And is he obligated to search for and remove them for fear of their desecration?

It is not permissible to discard items containing the name of Allah, a Qur'anic verse, a Prophetic Tradition (Hadith), or anything related to religion in unclean places, as this constitutes a desecration of them.

However, since you did this when you were young, there is no sin upon you, Allah willing, due to the saying of the Prophet (peace be upon him): "The Pen has been lifted from three: from the sleeper until he awakens, from the child until he reaches puberty, and from the insane until he regains his sanity."

You are not required to search for them now, due to the hardship and difficulty involved, as Allah Almighty says: "Allah does not intend to make any difficulty for you," and His saying: "And He has not placed upon you any difficulty in religion," and the saying of the Prophet (peace be upon him): "Indeed, religion is ease, and no one makes religion difficult except that it overwhelms him. So be firm, draw near, and give glad tidings."