For a general number field, this is not always true: see [1], which (among studying the phenomenon in general), gives the example of $E:y^2 = x^3+\frac54 x^2 -2x-7$ over $\mathbb Q(\sqrt[6]{-11})$ as an example of a curve for which $E_D$ has negative root number for every D. Assuming BSD, this would mean that the rank of $E_D$ is never 0.
The more general question of the distribution of ranks of quadratic twists over arbitrary number fields has been studied by Klagsbrun–Mazur–Rubin [2]: see, for example, Conj. 7.12, which gives a precise conjecture for the average rank of $E_D$ (as $D$ varies over $K^\times/K^{\times 2}$), generalising Goldfeld's conjecture. I'm not sure if examples such as the one above exist when $K = \mathbb Q(\sqrt{l_1}, \ldots,\sqrt{l_n})$, but this paper would be a good place to start.
On the other hand, a positive answer to your question is well-known when $K=\mathbb Q$: the existence of a single $D$ was first proven in [3] on the analytic side (see [4] for a proof for infinitely many $D$), and the algebraic side follows from Wiles' modularity theorem + Gross–Zagier's proof of rank 0 BSD. More recently, the work of Alex Smith has proven Goldfeld's conjecture almost unconditionally.
For general number fields, if you insist on only considering $E_D$ where $D$ is an integer, than I'm not really sure what the answer should be: the set of $E_D$ is an extremely sparse subset of the set of quadratic twists, where $D$ should vary over $K^\times/K^{\times 2}$.
On the other hand, if you're happy to vary $D$ over $K^\times/K^{\times 2}$, then Smith's methods almost certainly extend to general number fields as well, at least to some extent. So I'd imagine that, in cases where your statement is true, a proof is probably within reach.
[1] T. Dokchitser, V. Dokchitser, Elliptic curves with all quadratic twists of positive rank. Acta
Arith. 137 (2009)
[2] Klagsbrun, Zev, Barry Mazur, and Karl Rubin. "Disparity in Selmer ranks of quadratic twists of elliptic curves." Annals of Mathematics (2013): 287-320.
[3] Bump, Daniel, Solomon Friedberg, and Jeffrey Hoffstein. "Nonvanishing theorems for L-functions of modular." Invent. math 102 (1990): 543-618.
[4] K. Ono and C. Skinner, ‘Non-vanishing of quadratic twists of modular 𝐿-functions’, Invent. Math. 134(3) (1998),
651–660.
Best Answer
There is the paper "A Weil Theorem for Singular Curves" by Y Aubry and M Perret in the monograph Arithmetic Geometry and Coding Theory, Pelikaan et al, de Gruyter, 1996.
If you google by title you can find a scan of the paper. I am not an expert in this but it seems that $$ | 1+q-\#E(\mathbb{F}_q) | \leq 2 g \sqrt{q}+\Delta_E\leq 2 g \sqrt{q}+\pi-g\leq2 \pi \sqrt{g} $$
where $\pi$ is the arithmetic genus and $g$ the geometric genus of the curve, $$ \Delta_x=\#\{\tilde{E}(\mathbb{F}_q)\setminus E(\mathbb{F}_q)\}, $$ and $\tilde{E}$ is the normalization of the curve $E$ with an explicit product of cyclotomic polynomials.
Also, Bach, E. (1996). "Weil bounds for singular curves", Applicable Algebra in Engineering, Communication and Computing 7(4), 289–298. doi:10.1007/bf01195534