This is sometimes referred to as the "sine problem" in parallel with the much better known "cosine problem" of Chowla. I've seen the problem attributed to Bohr in connection with a problem about Dirichlet Series, but I've never seen a reference for that and I am not aware of what the application is. (Update: the paper of Konyagin cites the problem to Bohr's paper [3])
To state what is known, define
$S(K):= \min_{\{n_k\}_{k\in[K]}} \max_{\theta} |\sum_{1\leq k \leq K} \sin(n_k\theta) |$
where $\{n_k\}$ is an arbitrary set of $K$ natural numbers. Konyagin [1] has shown that:
$S(K) \gg K^{1/2} \left( \frac{\log K}{ \log \log K}\right)^{1/2}$
which gives a negative answer to the question as posed. In the other direction, Bourgain [2] constructed examples showing that $S(K) \ll K^{1/2 +1/6}.$$S(K) \ll K^{1/2 +1/6} = K^{2/3}.$
[1] S. V. Konyagin, Estimates of maxima of sine sums, East J. Approx. 3 (1997), no. 1, 63–7
[2] J. Bourgain, Sur les sommes de sinus, Harmonic analysis: study group on translationinvariant Banach spaces, Exp. No. 3, 9 pp., Publ. Math. Orsay 83, 1, Univ. Paris XI, Orsay, 1983.
[3] H. Bohr, A study of the uniform convergence of Dirichlet series and its connection with a problem concerning ordinary polynomails, Fys. Sall. Lund Forth 21 (1951) 103-118.