For example, let's say we have a straight, diagonal line $C_1$ connecting the points $(2,0)$ and $(0,1)$. Now consider a straight line $C_2$ that connects $(2,0)$ and $(0,0)$ and a straight line $C_3$ that connects $(0,0)$ and $(0,1)$
$C_1=C_2+C_3$, I think (also it's possible to choose parameterizations for all three lines such that the net direction of $C_1$ and $C_2$ is the direction of $C_3$, in case the lines are oriented)
So does the line integral of $C_1$ (i.e. $\int_{C_1} \vec F \cdot d\vec r$) equal the sum of the line integrals for $C_2$ and $C_3$?
What do you mean by the operation $C_2+C_3$? Is it the concatenation? But in this case only the endpoints are the same as $C_1$, and of course not the whole curve.
Always taking $+$ as concatenation, it is in general false what you asked: take for instance some $F$ that is equal to zero $(0,0,0)$ in a neighbourhood of $C_1$ and equal to $(1,1,1)$ in a neighbourhood of $C_2$ and $C_3$ (with smooth change for one value to another where the neighbourhood overlaps).