No, you definitely shouldn't, and yes, it is probably an infringement of copyright. Superimposing personal routes on a map (whether those routes are published in a book or not) is exactly what sites like [umap][1] are intended for.
But if you do decide to map these routes (which you shouldn't), please, please, **make sure that you clearly tag them as unsignposted**. Otherwise routers will say "turn onto Frodo's Hiking Route" when there are no signs for the walker to follow.
(The fact that only 300 ways [only 2 relations in the whole planet planet][2] are tagged as `unsigned=` perhaps tells you something about the wisdom of tagging such routes.)
[1]: http://umap.openstreetmap.fr/http://umap.openstreetmap.fr/
[2]: http://taginfo.openstreetmap.org/keys/?key=unsigned