Let $\alpha$ be a curve on a surface with Gaussian curvature $K=-1$ and let $N$ be the surface normal along $\alpha$. Assume that \begin{equation} \langle\alpha'(t),N(t)\rangle=0\quad\text{and}\quad\langle\alpha'(t),N'(t)\rangle\neq 0 \end{equation} for all $t$. I came across a document where the author says that if this holds then
$\alpha'(t)$ and $N'(t)$ are everywhere or nowhere parallel, i.e., $\alpha$ is everywhere or nowhere tangent to a principal direction.
I don't seem to know how to use the two given assumptions to prove that $\alpha'(t)$ and $N'(t)$ are everywhere or nowhere parallel, nor that $\alpha$ is everywhere or nowhere tangent to a principal direction. Actually, it is not clear to me why these two conclusions are equivalent: if for each $t$ we have that $\alpha$ is tangent to a principal direction, then $\alpha$ is a line of curvature. This means, using Rodrigues' curvature formula, that $N'(t)+k_i(t)\,\alpha'(t)=0$, where $k_i(t)$ is the principal curvature associated with the direction to which $\alpha$ is tangent. Then, N'(t) is everywhere parallel to $\alpha'(t)$ provided the $k_i(t)$ are constant, which is not the case in general.
Any idea regarding this proof of why the staments are not correct is really appreciated.
This seems to be false. What is correct is that $\alpha$ is never tangent to an asymptotic direction. The curve can certainly vary from a line of curvature without ever being tangent to an asymptotic direction. (The first condition is totally redundant.)