This was discussed a little on the algebraic topology list last autumn, you can look up the archives to see what was said.
Technologically, this is easy. The problems come in when you think beyond that.
How would such a site start? Initially, there would be very few reviews so no one would have a reason to visit the site (the probability of the paper wanted being reviewed being almost nil). For obvious reasons, importing an initial dataset from MathSciNet or Zentralblatt is extremely unlikely. If no-one visits, no-one's going to contribute.
How would such a site maintain itself? The big problem with reviews is that the person writing the review gets almost no gain from it but (to do it properly) has to put in a fair amount of effort (which is why so many reviews on MathSciNet and Zentralblatt are so appalling, just copying out the summary of the paper, and why the few gems are so greatly appreciated). That's a huge imbalance. MathSciNet sorts this out by awarding "points" to reviewers ("And points mean prizes!"). What would a free alternative offer?
How would such a site maintain its standards? Here, one gets into extremely murky waters. One idea suggested on the alg-top list was to use something like the stackoverflow model, but as this is going to involve opinions it could be extremely dangerous. Often, the most useful information in a review is a reason not to read a given paper - if you're looking at the review, you are probably already inclined to read it - and some of the classic reviews are those that rip apart a paper. Who's going to write those on an open system?
There are other ways of essentially filling the same role as reviews. You read a review to find out whether or not it's worth reading a paper. But the main problem comes before that, which papers should I read? Once I've found that, then I use the review more to figure out whether or not I should bother finding the paper in my library or not. If the paper is freely available, it's almost as quick to read the paper itself as to read the review (but I often read the review first because I find the paper via MathSciNet so the review is there anyway). So I would much prefer something to speed up finding papers. A better system of linking papers together: "If you enjoyed this paper, you might also enjoy ..." sort of thing.
So a really useful thing to do would be to have a "related papers" section linked to a given paper. This could be started by the author - who would have every incentive to do so (since it increases the likelihood that their paper is understood) and who would find it very easy to do (since they would have such a list of papers lying on their desk - it's all the articles, books, and so forth that they read when writing the paper in the first place). Basically, it's an expanded and commented reference list, and one which can be added to by others (so that if reading a paper, you find some other paper very useful which the original author didn't know about, or knew so well they didn't think to mention it, then you can add it).
Added later in response to some of the comments and the changes in the question.
First, a minor point. MathSciNet and the arXiv already have the capability to link articles together. If you look at a typical MathSciNet review then you'll see at the top right corner a box linking to where the article was cited, either in articles or reviews. I've found this an invaluable tool and I hope that the AMS will extend it historically (the links are only to recent articles where this data could be found fairly automatically). The arXiv has experimental support for full text search, so you can search for the arXiv identifier of one of your articles and find all those that cite it, for example.
Now on to my major point. "Someone should set up a site that ...". Who's to say that this has to be centralised? After the discussions on the alg-top list and the subsequent discussions on the rForum, I've come to the conclusion that having a central system isn't the best idea. All that is really needed is a central place from which all other places can be reached. And that already exists. It's called the arXiv. The arXiv accepts trackbacks (subject to some approval) so when you blog about a paper, send a trackback to the arXiv and then you should get linked to from the page on the paper.
Of course, this only works for articles that are on the arXiv. So then lobby the AMS to accept trackbacks as well (they can bung their standard "the AMS has no responsibility for non-AMS sites" disclaimer on). The basic information in MathSciNet is freely available, these trackbacks can easily be added to those.
In the meantime, instead of waiting for someone to come along with some central setup (which probably won't be quite what you personally were thinking of) simply put your notes online. Here's the message from the front page of the nLab:
We all make notes as we read papers, read books and doodle on pads of paper. The nLab is somewhere to put all those notes, and, incidentally, to make them available to others. Others might read them and add or polish them. But even if they don’t, it is still easier to link from them to other notes that you’ve made.
On the nLab you will find information about papers that people have found useful. You can search for a particular paper to see if anyone's commented on it, use it, or cited it. Some papers/books have their own pages. Here's one example http://ncatlab.org/nlab/show/Topological+Quantum+Field+Theories+from+Compact+Lie+Groups and here's another http://ncatlab.org/nlab/show/Elephant.
The basic point is that you can do this yourselves, now, without needing someone else to say "Here's the way to do it". You benefit right now without doing loads of extra work, and everyone else benefits incidentally as a result. Everyone wins. It doesn't have to be the nLab, you can use whatever software you like. Just put it online. Somewhere. Anywhere. Stick the arXiv/MR identifier somewhere prominent and the search engines will pick it up.
Then the rest of us will get into the habit of searching on the internet for articles of articles and find your comments.
(Incidentally, along with this, make sure that you put your own articles on your webpages. Every journal that I've ever encountered allows you to do this and this is really a Must-Do for academics. Even if you just put a scan of older papers, it's invaluable for those whose libraries don't carry subscriptions to every single journal under the sun (and those that have no library at all). There is No Excuse for not doing this, especially given that photocopiers now can easily scan straight to PDF.)