« de » only changes if a definite article is required to follow «de» directly - if le/la/l'/les are indicated, du/de la/de l'/des are used. 

This is uncommon with avoir envie de/besoin de, as these are usually referring to an unspecified 'quantity/amount' of something - which would not take a definite article (Aurelie notes that there can be exceptions). 

With « avoir peur de », the fear is usually 'general', or can be specific - in both cases definite article is required (general) or may be used (specific). 

It is fairly meaningless to say in English "I have fear of (some) cats" and the same is true in French:

 " I have fear of (all) cats" « J'ai peur (de les) des chats », or

 "I have fear of our cats/the black cat". ( J'ai peur de nos chats/ J'ai peur du chat noir)