Bratteli–Elliott–Jorgensen prove a range of classification results for unbounded derivations on a totally irrational noncommutative torus $C(\mathbb{T}^n_\theta)$, which basically say that any reasonable $\ast$-derivation will be the sum of a $\mathbb{R}$-linear combination of the infinitesimal generators $\delta_1,\dotsc,\delta_n$ of the translation action of $\mathbb{T}^n$ and an (approximately) inner derivation.
Here’s a typical special case, which follows from a combination of results in Bratteli–Elliott–Jorgensen. Let $\theta \in \mathbb{R}$ be irrational, let $U$ and $V$ be the usual unitary generators of the continuous noncommutative $2$-torus $C(\mathbb{T}^2_\theta)$, and let $\mathcal{O}(\mathbb{T}^2_\theta)$ be the dense $\ast$-subalgebra of Fourier polynomials in $U$, $V$. Then any $\ast$-derivation $\delta : \mathcal{O}(\mathbb{T}^2_\theta) \to \mathcal{O}(\mathbb{T}^2_\theta)$ has a unique decomposition $$\delta = c_1 \delta_1 + c_2 \delta_2 + \tilde{\delta},$$ where $c_1,c_2 \in \mathbb{R}$ and $\tilde{\delta}$ is inner, i.e., there exists self-adjoint $H \in \mathcal{O}(\mathbb{T}^2_\theta)$, such that $\tilde{\delta}(a) = i[H,a]$ for all $a \in \mathcal{O}(\mathbb{T}^2_\theta)$.
Please note that your would-be derivation $\delta$ is ill-defined, since you’d have
$$
\delta(VU) = 1 - (e^{2\pi i\theta})^3U^2 V^2, \quad \delta(e^{2\pi i \theta} UV) = e^{2\pi i \theta} - (e^{2\pi i \theta})^2 U^2V^2.
$$