How to get rid of or reduce the gap of an equation break?
I'd put the right block in an aligned environment with the [t] optional parameter, such that the top baselines are aligned. Note that with your current setup the spacing of the binary operators is wrong. I prefer putting the signs on the following line, though that might be a matter of taste.
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{align*}
A_C\Delta\phi_{2\,C}^\star - \sum_{i\in\mathcal{N}}A_i\Delta\phi_{2\,i}^\star = b_{C} +& A_C^x(\phi_{1\,C}^\star - \phi_{2\,C}^\star) - \\
& A_W(\phi_{1\,W}^\star - \phi_{2\,W}^\star) - \\
& A_E(\phi_{1\,E}^\star - \phi_{2\,E}^\star),
\end{align*}
\[
A_C\Delta\phi_{2\,C}^\star - \sum_{i\in\mathcal{N}}A_i\Delta\phi_{2\,i}^\star =
\begin{aligned}[t]
b_C &+ A_C^x(\phi_{1\,C}^\star - \phi_{2\,C}^\star) \\
&- A_W(\phi_{1\,W}^\star - \phi_{2\,W}^\star) \\
&-A_E(\phi_{1\,E}^\star - \phi_{2\,E}^\star),
\end{aligned}
\]
\end{document}
Simply put the \sum in a \smash[b]{...} command. I took the opportunity to propose a slightly different layout for this equation:
\documentclass{article}
\usepackage{amsmath, amssymb}
\begin{document}
\begin{align*}
A_C\Delta\phi_{2\,C}^\star -\smash[b]{\sum_{i\in\mathcal{N}}}A_i\Delta\phi_{2\,i}^\star = b_{C} +{}& A_C^x(\phi_{1\,C}^\star - \phi_{2\,C}^\star) - \\
& A_W(\phi_{1\,W}^\star - \phi_{2\,W}^\star) - \\
& A_E(\phi_{1\,E}^\star - \phi_{2\,E}^\star),
\end{align*}
\begin{align*}
A_C\Delta\phi_{2\,C}^\star -\smash[b]{\sum_{i\in\mathcal{N}}}A_i\Delta\phi_{2\,i}^\star = b_{C} & +A_C^x(\phi_{1\,C}^\star - \phi_{2\,C}^\star) \\
& -A_W(\phi_{1\,W}^\star - \phi_{2\,W}^\star) \\
&- A_E(\phi_{1\,E}^\star - \phi_{2\,E}^\star),
\end{align*}
\end{document}
\begin{align*}
A_C\Delta\phi_{2\,C}^\star - \sum_{i\in\mathcal{N}}A_i\Delta\phi_{2\,i}^\star = b_{C} &+ A_C^x(\phi_{1\,C}^\star - \phi_{2\,C}^\star) \\[-.75pc]
&- A_W(\phi_{1\,W}^\star - \phi_{2\,W}^\star) \\
&- A_E(\phi_{1\,E}^\star - \phi_{2\,E}^\star),
\end{align*}
Try out this!