Exact exponential-time algorithms for 0-1 programs with nonnegative data
if the number of non-zero coefficients in $A$ is linear in $n$, there is an algorithm that solves this problem in less than $2^n$ time.

Here's how it works.  We use the standard connection between an optimization problem and its corresponding decision problem.  To test whether there exists a solution $x$ where $Ax\le b$ and $c^T x \ge \alpha$, we will form a decision problem: we will adjoin the constraint $c^T x\ge \alpha$ to the matrix $A$, and test whether there exists any $x$ such that $Ax \le b$ and $-c^T x \le -\alpha$.  In particular, we will form a new matrix $A'$ by taking $A$ and adding an extra row containing $-c^T$, and we will form $b'$ by taking $b$ and adjoining an extra row with $-\alpha$.  We obtain a decision problem: does there exist $x \in \{0,1\}^n$ such that $A' x \le b'$?  The answer to this decision problem tells us whether there exists a solution to the original optimization problem of value $\alpha$ or greater.  Moreover, as explained in the answer to your prior question, this decision problem can be solved in less than $2^n$ time, if the number of non-zero coefficients in $A'$ is linear in $n$ (and thus if the number of non-zero coefficients in $A$ is linear in $n$).  Now we can use binary search on $\alpha$ to solve your optimization problem in less than $2^n$ time.

My thanks to AustinBuchanan and Stefan Schneider for helping to debug an earlier version of this answer.