As the VOR frequency is automatically selected when the DME is selected, under what system conditions is manual tuning of the VOR required?
As pointed out in the comments, I think you have things turned around. If you select the frequency of a VOR that has a co-located DME, the system will automatically also select the DME frequency. Thus, I'm guessing that what you perhaps meant to ask is under what conditions do you manually tune the DME frequency. If that is NOT what you are asking, put a comment after this answer to that effect, and I will delete this answer.
Now, if that is what you are asking, the answer in one sense is "never" in that, at least in civilian aircraft, you always select the VOR frequency. However, prior to my retirement in 1999 there were a few, very few as I remember, facilities that provided DME but at which there was no co-located VOR. To get the DME, you selected the VOR frequency that would have been paired with the DME had there been a co-located VOR. and it was the frequency of that non-existent VOR that was shown on the charts along with a clear and obvious note that there was no actual VOR. I have no idea whether in our current airspace system such still facilities exist.