If the arXiv had referees, would conventional journal publications still make sense? In the last decade, more and more researchers read the scientific literature on electronic copies or on printed copies, but not anymore directly from the printed journal. 
Nowadays, I think the most important role of journals is therefore only to offer a well-established editorial and refereeing platform.

Anyway, there is a sort of economic loophole. Let me explain.
Research institutions pay both researchers and journals (to have online access and for printed copies) where researchers publish. On the other hand, journals offer a refereeing platform which is based on the unpaid work of referees, which are, again, researchers. Therefore, research institutions pay for all, and journals earn money for the research published. I could be a little naive here, but I think not far from truth. 

Therefore, why are journals useful, apart from refereeing?
For instance, imagine that arXiv, or another open-access archive will introduce refereeing and some sort of editorial filtering. In this scenario, will conventional journal publications still make sense?
I used to think in a way similar to the question, but I had a meaningful conversation with a prestigious scholar one time in graduate school, who pointed out that I was missing a key purpose of journals: permanent archival.

The arXiv has been around for about 25 years. We have no idea whether the papers on it will be readable in 100 years. But we know that we can read documents on paper from ancient Egypt, and we have every reason to  believe that journals printed on paper and stored in archival libraries can last just as long.  

The scholar I spoke with was not particularly worried about refereeing. He was worried about whether his work would still be available in 1000 years. Of course, I didn't ask him if he thought anyone would still be interested in it :)