However, there will be app features that depends on permissions to work properly. Below are what I've found from experimenting.
1. Contacts - If you deny permission to contacts, all your chats will show up as a list of phone numbers with no names (You can always view the profile picture and WhatsApp username is someone's profile, if they allow).
2. Location - If your location services are off, the functionality to share live location will not work. Sharing a location from Google Maps (you can always search this for places with location off) will still work.
3. Storage - You will not be able to post pictures into chats, download pictures from chats, post or view a WhatsApp status.
4. Microphone - Calls and recordings will not work
5. Camera - Video calls will not work, nor will the feature in status/chats to snap a picture and share instantly