Should I refrigerate homemade yogurt before making Greek yogurt?
It isn't necessary, and many recipes don't call for it. 

But the results will be slightly different, since the yogurt will continue to ferment (even in the fridge), and the gel structure of the proteins will set. That will alter composition of the whey that you're straining out, which means the resulting difference will have a slightly different texture and flavor.

Since that texture and flavor also depend on the milk you use, the temperature you process it at, the strain of culture you use, etc. I can't really say exactly what the differences will be, and how significant they are. You'd have to make it yourself and see -- and, of course, decide which you prefer.

I can say, however, that you won't ruin a batch of yogurt by straining before refrigerating.