Set Theory Properties of the Budget Constraint In Microeconomic theory, the budget constraint is defined by 4 distinct properties:
It is


Bounded
Closed
Convex
Non-empty


The 1. 2. and 4. are very straight forward and the benefits in terms of analysis are apparent. My question is with respect to convexity.

Convexity of a set implies that any line between two points is the set is also within the set (i.e. take any two points say $q$ and $q'$ that both satisfy the constraint, then any linear combination of the $q$ and $q'$ also satisfy the constraint)

My question is, what is the benefit or added utility to this property holding?
I immediately think of a budget constraint that, rather than exhibiting diminishing marginal returns, exhibits increases returns in a kind of learning sense: As more of one good is produced, the production becomes more efficient as the producer is learning. Whether or not this is a realistic circumstance, it begs the question.

What happens if convexity is violated? or rather, what can we no longer do?
Convexity can be a very important assumption as much of economic analysis is built on working with convex sets, which makes things easier. 

Very importantly, convexity of sets allows us to work with the Separating Hyperplane and Supporting Hyperplane theorems, which have applications to many results in economics, as partially discussed here and here.

Here is also a short, concrete example of something for which we need convexity of the budget set:

Suppose you want to prove that Marshallian Demand $x(p,m)$ is single-valued for every (larger than zero) price ($p$) and (non-negative) income ($m$) combination. That is to say, a Marshallian Demand Function exists, since without single-valuedness we would have a Marshallian Demand Correspondence (i.e. we'd have multiple good combinations as part of the optimal demand choice for a single given price and income level), which can be harder to work with. This is a pretty nice result that we (almost) always work with in undergraduate texts. For that you need strictly convex preferences and a convex budget set $B(p,m)$. The proof is as follows:

If two points $x_1$ and $x_2$ are both elements of the demand $x(p,m)$ at given prices $p$ and income $m$, i.e. if $x_1, x_2 \in x(p,m)$, then by definition we must have indifference between the two points, that is $x_1 \sim x_2$. For a single-valued demand function two different points $x_1$ and $x_2$ cannot exist  for single given level of prices and income, i.e. they would have to be the same point. In other words  $x_1 = x_2$. We prove it by contradiction. Suppose $x_1 \neq x_2$.

For all $\lambda \in (0,1)$, we have $\lambda *x_1 + (1-\lambda) *x_2 \in B(p,m)$ by convexity of $B$. By strict convexity of preferences we have $\lambda *x_1 + (1-\lambda) *x_2 \succ x_1$. However, this contradicts the fact that $x_1 \in x(p,m)$ in the first place. Hence, $x_1 = x_2$.

So, in words, some combination of $x_1$ and $x_2$ is better than just $x_1$ at the given price and income level. But such a combination is also attainable at that given price and income level because of the convexity of the budget set. This is where the budget set convexity assumption is important. Furthermore, if for a given price and income level there is a point (in this case a combination of $x_1$ and $x_2$) that is both strictly better and attainable, then this contradicts the original starting point that $x_1$ is a point in the demand function for the given price and income level.

As a side note, I think it is a bit misleading to talk about producers in your post when talking about budget constraints. If we are looking at a budget constraint, we should be discussing consumers.