OP asks:

What is the sign (+tive/-tive) of the third order central moment of a positive linear combination of log-normal random variables`

Let $Z = a X + b Y$. Then, for any random variables $X$ and $Y$ that are *independent*, it can be shown that:

$$\mu_3(Z) = a^3 \mu_3(X) + b^3 \mu_3(Y)$$

where $\mu_3(W) = E[(W-\acute\mu_W)^3]$ denotes the third central moment of any random variable $W$ whose moments exist, and $\acute\mu_W = E[W]$.

If $X$ is a Lognormal random variable, then:

$$\mu_3(X) = \left(e^{\sigma ^2}-1\right)^2 \left(e^{\sigma ^2}+2\right) e^{3 \mu +\frac{3 \sigma ^2}{2}}$$

is strictly positive, and similarly for $Y$.

Thus, if $a$ and $b$ are positive, it follows that $\mu_3(Z) = E[(Z-\acute\mu_Z)^3]$ is positive.