Is the following series consisting of equally distributed $\pm 1$ bounded?

The sequence $\sum a_n$ is unbounded.

This is a consequence of a general result from Kesten,
On a conjecture of Erdös and Szüsz related to uniform distribution mod 1, Acta Arithmetica (1966). The proof is not very long but quite computational, using properties of continued fractions.

Theorem Let $\xi \in [0,1]$, $0\leq a < b \leq 1$, denote by $N(M, \xi, a,b)$ the number of integers less than $M$ such that $a \leq \{k\xi\} \leq b$. Then $N(M,\xi, a,b) - M(b-a)$ is unbounded if $b-a$ is rational and $\xi$ is irrational.

You are interested in the $1/2$ discrepancy of an irrational rotation of the circle. Note that from the Denjoy-Koksma inequality, there is a sequence $q_n$ such that $\sum_{k= 0}^{q_n-1} a_k$ is bounded by $2$. The sequence exists for any irrational rotation of the circle and is given by the denominators of the partial fraction decomposition of the rotation number. For the golden ratio, these numbers are given by the Fibonacci numbers.

The golden ratio is a bit special because its decomposition in continued fraction is just $[1,1,1...]$ so it is badly approximated by the rational numbers. So the computation of Kesten should be simpler in that case.

EDIT: the theorem is also mentioned shortly in the book of Kuipers and Niederreiter, "uniform distribution of sequences". Some additional references can be found in the notes of chapter 2, section 3 p128.

I've decided to upgrade my comments and make an answer out of them, even though I'm just addressing the (easier) variant suggested by the OP at the end of the post, where we replace the golden ratio by other numbers $b$.

If $b$ can be well approximated by rationals, then it's easy to see that the sum cannot stay bounded: More precisely, suppose there are infinitely many (reduced) fractions with odd denominators $q$ such that $$ \left| \frac{b}{2} - \frac{p}{q} \right| = o(q^{-2}) $$ (the golden ratio is not of this type; I can only get $O(q^{-2})$, and in fact it is usually thought of as the number with the worst rational approximations).

The shift by $p/q$ is periodic with odd period, so gives a bias of at least $1$ to one of the two halves of the circle. Also, if I keep a distance $\gtrsim 1/q$ to the endpoints of the intervals, then a perturbation by $\le \delta/q$ of the initial point will not change anything.

Now I can just wait until $nb$ is close to such an initial point, and the approximation of $b/2$ by $p/q$ will be justified during a time interval of length $kq$, for any $k\ge 1$ (because $kq\cdot o(q^{-2})\ll 1/q$). So I pick up an overall bias of at least $k$.