= Solution
Keep the <Electron> number fixed, so
$$
N_e=\int_0^\infty g(\epsilon)p_F(\epsilon)\,d\epsilon
$$
and $\mu(0)=E_F$. Heating smooths the <Fermi-Dirac distribution> across the <Fermi energy>, creating holes below $\mu$ and particles above it. The sign of the low-<temperature> adjustment of <chemical potential> follows from the local slope of the <density of states>.
In one dimension, $g\propto\epsilon^{-1/2}$ decreases with <energy>. At fixed $\mu$, smoothing removes more <Electrons> below it than it adds above it, so $\mu$ initially rises. In three dimensions, $g\propto\epsilon^{1/2}$ increases, so $\mu$ falls. In two dimensions the constant <density of states> makes the leading changes cancel, so $\mu$ stays extremely close to $E_F$ in the degenerate regime.
For quantitative checks, a <Sommerfeld expansion> gives, for $k_BT\ll E_F$,
$$
\boxed{\mu_1(T)=E_F+\frac{\pi^2}{12}\frac{(k_BT)^2}{E_F}+\cdots,\qquad
\mu_3(T)=E_F-\frac{\pi^2}{12}\frac{(k_BT)^2}{E_F}+\cdots.}
$$
For a constant two-dimensional <density of states> above an unbounded parabolic band bottom, the number <integral> can be done exactly:
$$
N_e=g_2k_BT\ln(1+e^{\mu_2/(k_BT)})=g_2E_F,
$$
so
$$
\boxed{\mu_2(T)=k_BT\ln\left(e^{E_F/(k_BT)}-1\right)
=E_F+k_BT\ln\left(1-e^{-E_F/(k_BT)}\right).}
$$
It actually decreases, but its low-<temperature> correction is exponentially small rather than quadratic. “Constant in two dimensions” is therefore a low-<temperature> approximation, not an exact statement for every $T$.
The one-dimensional rise is also only an initial trend. In the high-<temperature> <Maxwell-Boltzmann distribution> limit, the ideal $d$-dimensional <electron gas> has $\mu\simeq k_BT\ln[n_d\lambda_T^d/2]$, where $n_d=N_e/L^d$ and $\lambda_T=\sqrt{2\pi\hbar^2/(mk_BT)}$ is the <thermal de Broglie wavelength>. It eventually becomes negative and decreases in every dimension. The one-dimensional <chemical potential> thus passes through a maximum. This high-<temperature> statement refers to the stipulated unbounded free-electron <dispersion relation>, not a claim that a real finite-width crystal band remains parabolic indefinitely.
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