Question: For which unbounded domains $D$ does the condition $\liminf_{x\to y}u(x)\geq0$ for all $y$ in $\partial D$ (the finite boundary of $D$), imply that $ u$ in nonnegative in $D$ for subharmonic functions in $D$. 

Answer: The exact criterion is 

$(*)$ The point at infinity should have zero harmonic measure in $\partial^\infty D$, i.e., Brownian motion $W_t$ started at a point in $D$ should hit $\partial D$ almost surely. 

Indeed, if $(*)$ holds, then the required inequality follows from the supermartingale property of $u(W_t)$ for superharmonic $u$. And if $(*)$ does, not hold define $-u(x0$ to be the harmonic measure of the point at infinity for Brownian motion started at $x$. 

There are several other equivalent criteria:

(1) The Martin capacity of $\partial D \setminus B(0,R)$ should be be bounded away from zero as $R \to \infty$;

(2) A Wiener test in terms of Greenian capacity:

 Consider the shells  $S_k=B(0,2^k) \setminus B(0,2^{k-1})$. Then the requirement is 
$$\sum_{k \ge 1} 2^{k(2-d)} \cdot \text{Cap}_G (\partial D \cap S_k) =\infty \,.$$

[1] Benjamini, Itai, Robin Pemantle, and Yuval Peres. "Martin capacity for Markov chains." The Annals of Probability (1995): 1332-1346.

[2] Lamperti, John. "Wiener's test and Markov chains." Journal of Mathematical Analysis and Applications 6, no. 1 (1963): 58-66.