The answer is no, but it's actually a deep question and leads to another metric on the Teichmüller space of surfaces. The minimum quasi-conformal constant of a map in a given homotopy class is the Teichmüller metric on surfaces; it is also equal to the maximal ratio of extremal lengths of any curves. If you look at the ratio of hyperbolic lengths, as in your question, you get a quite different metric on the space of surfaces, studied by William Thurston: Minimal stretch maps between hyperbolic surfaces. That paper effectively uses an asymmetric version of the inequality ($\ell(f(C)) \le K \ell(C)$), but the symmetrized version is also a very different metric from the standard one given by quasiconformal stretch. Both the asymmetric stretch metric was studied recently in detail by Dumas, Lenzhen, Rafi, and Tao.
The paper The converse of the Schwarz Lemma is false by Maxime Fortier-Borque gives many relevant examples, although that paper is concerned with the slightly different situation of embeddings between non-compact surfaces and maps that decrease length, effectively the case $K=1$ in your questions.