I have some apps on my HTC One (M8) that want to have access to my camera and microphone (Yahoo Mail app). I have seen on an iOS device that you can disable this access but still run the app. Is there something similar in the Android world?
Not directly, but with a rooted device, Xposed or a custom ROM (e.g. CyanogenMod) you are able to revoke permissions granted to the app by Android.
If your device allows the installation of applications signed with developer keys, then you might be able to do that without root needed.
Side effects may be unexpected application crashes or hangs if the app decides to do something that needs one of the revoked permissions.
In the best case the application will just not be able to use e.g. the camera and show a black preview of the image (Camera App for example).