I'm afraid that this question might be a little vague, so I try to put in as many details as possible.
Proposition 6.7 of Hartshorne's Algebraic Geometry characterizes projective curves as being complete (i.e., proper) and so my question is really about what can be said of curves that fail to be proper.
In general, there are some conveniences when varieties are proper. For example (although perhaps a little far-fetched but they come off the back of my head as they are related to what I'm doing), given a proper variety $X$ defined over a number field $k$, we know that $X(\mathbb{A}_k) = X(k_\Omega)$ and so strong and weak approximations are the same (the former topological space is of great interest in the theory of rational points). Furthermore, we have $\bar{k}[X]^* = \bar{k}^*$ and this simplifies the Hochschild-Serre spectral sequence, say.
So back to my question, all of the abovementioned book assumes varieties to be projective (hence proper). Let's at least suppose that a curve is a smooth geometrically integral dimension 1 variety of finite type over $k$. Could we, say, talk about divisors on an open curve the same way? If so, are the notions of the divisor class group, the Picard group, etc, still the same? Would the Neron-Severi group of this curve still be $\mathbb{Z}$?
I would think that all of the answers to the questions would be yes, but what immediate issues (or inconveniences) would arise when we drop the proper condition on curves?