I know nothing about the Tao Te Ching, but the sentences you quote are not (logically) contradictory, or paradoxical:
The Tao that can be trodden is not the enduring and unchanging Tao.
Nothing paradoxical here, is there? Maybe the only enduring and unchanging Tao is one that people is prevented from trodding -- so as to avoid wear from all that trodding, and stay unchanged, say.
Suppose we introduce the following notation:
T: stands for the property of being a Tao
EU: stands for the property of being enduring and unchanging
R: stands for the property of being susceptible of being trodden
then your sentence can be rendered thus:
For all x, (Tx ^ Rx -> ~ EUx)
This is classical logic at its simplest.
[EDIT: As Mozibur suggests, one might want to argue that a path is, necessarily, susceptible of being trodden. I am not sure about that: consider a path inside a flooded mine, for instance. Anyway, if paths are to be practicable of necessity, then there is a contradiction in that sentence.]
The name that can be named is not the enduring and unchanging name.
Suppose that all names can be named. (Just like, for example, "Schiphol" is a name of my name.) If so, no name is enduring and unchanging. No paradox here either.
On the other hand, maybe there is an enduring and unchanging name, and that name has no name. This is semantically implausible, but not paradoxical.
Anyway, introducing the notation:
Nxy: stands for the relation of x being a name of y
We can say that a is a name iff there is an y such that Nay, and your second sentence can be rendered thus:
for all x (there is a y (Nxy) -> [there is z(Nzx) -> ~EUx])
Again classical, consistent logic.