I notice from your tags that you refer to Britain, and unlike many people on this site, I live there.
Firstly, there are certain phrases that are still commonly used in Britain, which may not be taken in a literal sense but are understood immediately by almost every British person. Some examples of these could be 'a rose by any other name', 'alas, poor Yorick', 'friends, Romans and countrymen', and, 'gone now is the winter of our discontent'. All these are Shakespearean and still used to create an immediate image, which is recognisable to the hearer, despite its use of ellipsis. This is why Shakespeare is still seen as relevant in todays society, even though we know that his plays and poetry is probably written by Christopher Marlowe (a conspiracy theory), that his plays are not overly trueful and heavily propagandised (Richard III was not really a club-footed, hunchback coward - no, he was something even worse, a Yorkshireman), and he was a plagiarist.
Secondly, some literature is quite simply a magnificent read, which stands the test of time. Homer's 'Iliad' is just such a poem. Only a couple of years ago, Hollywood bastardised the story, propagandised it am=nd modernised it, ruining the whole point. A truly great piece of work lives forever, whilst rubbish disappears. Surely there should be such a place for this kind of literature, too.
The only difference between a 'reading heritage' for a student in 2015 and one from 1915 would be the addition of a small number of modern classics - '1984' by George Orwell, 'Catch 22' by Joseph Heller, 'The Name of the Rose' by Umberto Eco and 'The Great Gatsby' by F. Scott Fitzgerald stand out to my mind. On a personal note, 'The Little Prince' by Antoine de St Exupery, 'The Hitch-hikers Guide to the Galaxy' by Douglas Adams and 'Blitzed: The Autobiography of Steve Strange' would also be included.
Beyond that, the usual golden oldies of Homer, Virgil, Tacitus, Chaucer, Cervantes, Hugo, Blake, Byron, Brecht, Pagnol, Miller & Pinter (both modernish), Donne & Johnson, Aristophenes and Dorthy Parker.
Most modern heritage is from tv and cinema, and whilst elements of that media may live on, they rarely exist in a textual state, so would not be in a 'reading heritage'. As for bloggers - well only time will tell, though I do not feel that many, if any, of todays bloggers or twitterers will prove to be a sustained force. By their nature, they will be more in keeping with Andy Warhols '15 minutes of fame' than a heritage. I might well be wrong, but blogs tend to be very much of that moment, and become out of date quickly. A blog may be popular one day, then old hat the next. A little like music, many youngsters will not want to read the same bloggers that there parents did, so these blogs will die out. The internet itself is rather ethereal, whereas a paper book stands alone and is tactile and more permanent.
I notice that an Italian chappie believes that people will no longer read books, yet one of the growth markets in the last year has been in the 'virtual novel' genre, with books downloadable and text-readers selling well.
Whatever does happen, Dan Brown has no place in any discussion on literature. Not only are his books the worst kind of abuse of the arts but absolute drivel. I am not religious, so don't think my religious beliefs are offended, it is my beliefs about good writing, accuracy of detail and a damn good story. He fails at every test. (I teach semiotics, as well as business English), I have to de-Dan Brown students and explain the genuine symbolism in art and literature. He doesn't even call his man a semiotologist, he calls him a symbologist!! Please, someone shoot him for me. (He's my current pet hate)