For an arithmetic progression of the form $a_i = ki, i \in \mathbb{N}$, the question is trivial - if $k$ is a power of 2, then the progression will generate an infinite number of powers of 2, and no powers of 2 for any other case.
I'm having trouble figuring out the more general $a_0 \neq 0 $ case. What form does the progression $a_0 + k\mathbb{N}$ have to take to generate an infinite number of powers of 2? I.e., if an arithmetic progression generates an infinite number of powers of 2, what will the relationship between $a_0$ and $k$ look like?
Fix a common difference $d$. Take the $d$ arithmetic progressions $A_0,A_1,\ldots, A_{d-1}$ all with common difference $d$ with the first term for $A_i$ chosen as $i$. They partition the set of positive integers. As they are finite in number at least one of them should contain infinitely many powers of 2.