In another question about distance PhDs (Do any schools offer teaching or research assistantships via distance education?), I received several comments noting that PhDs on-line are of inferior quality to on-campus PhDs. If what is seen of PhD work from PHD Comics is accurate, PhD students seem to spend much time working independently. So, what aspect of a PhD cannot translate to the on-line format? What aspect of the process cannot be effectively conducted via E-mail or any of the other various forms of on-line communication?
A general answer is that a lot of PhD training is informal: it involves things that are taught by example, by osmosis, by noticing someone doing something wrong and correcting them (or right, and praising them). This kind of informal training is much more difficult in online interactions. Some examples are given below...
One specific aspect that does not translate well to online learning is the cross-pollination of ideas via proximity.
A big part of training PhD students involves putting a bunch of smart, talented, hardworking people together and letting them learn from, motivate, support, and bounce ideas off one another. A prerequisite for this is proximity; you just can't have the same kind of interactions over email. Some research centers are explicitly designed to encourage this. At the Bell Labs facility in Murray Hill, NJ,
Some of the hallways in the building were designed to be so long that to look down their length was to see the end disappear at a vanishing point. Traveling the hall’s length without encountering a number of acquaintances, problems, diversions and ideas was almost impossible.
(Source: New York Times)
The research center I currently work in was built with a long hallway with this in mind, specifically to foster collaboration between students in different groups!
Another thing that does not translate well is learning scientific communication and mentoring skills. As a PhD student, I have many chances to practice speaking about my work to other graduate students, professors, and undergraduates. I also get to mentor M.S., B.S., and high school students in my lab. These skills are much more difficult to learn over distance.
A third thing that does not work well over email and Skype is learning the etiquette and standards of the field. This is something that isn't explicitly taught (usually) but that students are expected to pick up by osmosis from spending time with other academics and others in their field.
ff524 provides an excellent answer—although I think there are several other important points that are not discussed there.
Experimental work cannot easily be done via long-distance arrangements, unless one happens to be near another research institution with the necessary equipment and support staff to enable the research to take place.
It is also worth noting that online interactions are still not ideal for rapid development of new ideas and discussions. For instance, suppose during a conversation with your advisor, you want to make a quick sketch and show it to her. If you're meeting in person, you can easily develop the figure on paper or on a whiteboard (or similar), and adjust and make various comments. Such an exchange is extremely tedious online—to the point of being almost counterproductive.
This is meant to be in addition to your other answers, so will refer to them a bit.
First let's take your clarifying comment: "For non-STEM fields" I work in a STEM field, but have spent time discussing related issues with postgrads working across the university. Many of those in non-STEM fields seem or feel rather isolated anyway compared to those of use in a research group which works closely together. This is partly due to the organisational structure and funding situation leading to fewer PhD positions in some fields, as I understand it, both of which will vary between universities and countries. This could go either way - it could mean that remote working takes away the last vestiges of academic human contact, or it could mean that your tucked away in a corner of your own home alone, rather than a corner of the office. This relates to ff524's answer as well, though I'm coming at it more from the important aspect of getting through a rather strange life for a few years rather than learning academic or transferrable skills.
How much library research (wrt aeismail's answer) is necessary offline will vary hugely with subject - in my case very little, though access to textbooks at all but especially the early stages must not be neglected. They may not be realistically available for you to buy. Access to a more local university library as a visiting reader may be possible, but it would be unwise to rely on this and you may not be able to borrow the books.
You need a supervisor who's not just willing, but enthusiastic about the idea. I wouldn't want to be a supervisor's first remote student - let someone else iron out the difficulties first. You need to be on the same wavelength when it comes to collaboration tools, which means in practice you need to be happy to work with the tools your supervisor wants to use. This is a minor issue when you work in the same building, but becomes critical when you're widely separated.
I would like to stress a bit more the role of "motivation" as in the answers of ff524 and Chris H., specially in the emotional aspect of it. I would say that doing a PhD has an implicit feeling of isolation: in general, you become an expert at a really narrow field compared to the whole human knowledge (I love how Matt Might explains it in his Illustrated guide to a Ph.D.) and that makes sharing your experiences/thoughts harder (how would you share your troubles and toils with your girlfriend?)
Even if there is no one in your research group doing anything related, the fact of sharing a space (not even talking!) with other people going through the same processes is comforting somehow.
Intuitively, adding a layer of physical isolation on top of the experience would make it less stimulant.