Why should we not store onions and potatoes together?
As you can see at the Chapter 7: Storage of horticultural crops of the book PDF: Small-Scale Postharvest Handling Practices: A Manual for Horticultural Crops (4th Edition):


If you look p.166 (p.173 on the PDF), you can see that the onions need to be stored in this conditions:

between 0-5°C with a relative humidity (R.H.) of 65-70%

If you look p.168 (p.175 on the PDF), you can see that the potatoes have different needs:

Fresh market: between 4-7°C
Processing: between 8-12°C
seed potatoes: between 0-2°C
But must of all, potatoes need very highest relative humidity (R.H.): 95-98% versus 65-70% for onions



So, if you store them together, either one or the other will be in bad storage conditions.

Further, as you can see at Section 7: Storage of horticultural crops of Small-scale postharvest handling practices - A manual for horticultural crops - 3rd edition:


Dry onions (Group 3) are very sensitive to moisture, which makes their storage incompatible with the high R.H. needed for potatoes. 
New potatoes (Group 6) may also produce ethylene, which will badly damage green onions (Group 2) if stored together.


Anyway, storage of horticultural crops cannot just be resumed by "don't put this or that together", as you may have seen:
it is important to consider R.H., temperature, and many other factors.

If the subject interests you, all this (and many other things) is pretty well explained in the following references:


PDF: Small-Scale Postharvest Handling Practices: A Manual for Horticultural Crops (4th Edition)
Web: Small-Scale Postharvest Handling Practices: A Manual for Horticultural Crops (4th Edition)
WEB: Small-scale postharvest handling practices - A manual for horticultural crops - 3rd edition


Ethylene is also a "critical factor", and this topic is more developped in the following resources:


PDF: Postharvest Ethylene: A critical factor in quality management.
Book: Postharvest Technology of Horticultural Crops (see Ethylene in Postharvest Technology pp.149-162)