Does uncountable summation, with a finite sum, ever occur in mathematics? Obviously, “most” of the terms must cancel out with opposite algebraic sign.
You can contrive examples such as the sum of the members of R being 0, but does an uncountable sum, with a finite sum, ever occur naturally as part of a larger scenario (eg, proof of a theorem, or preparation for a definition)?

If $I$ is a set and $f:I\to\mathbb R$ is a function, there is a standard definition of what it means that $f$ be summable. 
One compact way of giving it is as follows. Let $\mathscr P_{\mathrm{fin}}(I)$ be the set of all finite subsets of $I$, and let $\bar f:\mathscr P_{\mathrm{fin}}(I)\to\mathbb R$ be function such that $$\bar f(A)=\sum_{a\in A}f(a)$$ for each $A\in\mathscr P_{\mathrm{fin}}(I)$.
The set $\mathscr P_{\mathrm{fin}}(I)$ is partially ordered by inclusion, and with this order it is directed, so we can view $\bar f:\mathscr P_{\mathrm{fin}}(I)\to\mathbb R$ as a net in $\mathbb R$.
We say that the function $f$ is summable if the net $\bar f$ converges, and the limit of the net is by definition the sum of $f$. This notion extends absolute summability of series; I am not aware of any sensible meaning of conditional summability of such a family -- so I don't think one deals at all with "cancellation".
Now, since $\mathbb R$ is a topological space with a denumerable basis, one can easily check that 

if a function $f:I\to\mathbb R$ is summable according to this definition, then the set $\{i\in I:f(i)\neq0\}$ is countable.

