Learn French: Que déplacez-vous de crainte de fissurer le verre ? - What are you moving for fear of cracking the glass?

We use the infinitive because the person doing the moving (vous) is the same person who might crack the glass. When the subject is the same for both actions, you use de crainte de followed by an infinitive. If the subjects were different, you would use de crainte que followed by the subjunctive.

Is there a difference between de crainte de and de peur de?

They mean exactly the same thing and both express a negative purpose, translating to for fear of doing something. However, de crainte de is slightly more formal and literary than de peur de.

Why is the question structured as Que déplacez-vous rather than using est-ce que?

This uses subject-verb inversion, which is a formal way to ask a question in French. You could absolutely say Qu'est-ce que vous déplacez, which is more common in everyday spoken French, but the inversion perfectly matches the slightly formal tone of de crainte de.

Could le verre refer to a drinking glass here, or just the material?

It can mean either! Le verre is the word for the material itself, but it is also exactly what you call a drinking glass. Depending on the broader context, this sentence could be about moving something heavy to avoid cracking a glass tabletop, or carefully shifting items to avoid breaking a drinking glass.