In addition to other problems described here, is there also a problem of hermiticity of the Hamiltonian operator $\hat{H}$, if someone erroneously writes $$\hat{H}:=i\hbar\frac{\partial}{\partial t}?$$
If yes, what is the origin of non-hermiticity? I guess this because $t$ is usually taken to be in the domain $0\leq t\leq +\infty$ whereas $-\infty\leq x,p\leq+\infty$. But I'm not sure because very often by the choice of origin $t$ can be made to lie in the domain $-\infty\leq t\leq+\infty$.
The real question, as others have noted, is what are you going to apply this operator to exactly? Let's just elaborate on that to show you what goes wrong.
Let $\Psi(x,t)$ be a generic solution to the 1-D Schrodinger Equation. The conventional interpretation of this solution is that at each time $t$, the single-variable function $$\psi_t(x):=\Psi(x,t)$$denotes the state of the system at time $t$, and the totality of such states form a Hilbert Space, replete with Hermitian operators and all that jazz.
But since these wavefunctions are functions of $x$, how do we apply an operator like $i\hbar\partial_t $ to them? For this we need functions of time $t$.
One obvious strategy to try is to simply switch $t$ and $x$ around in the above definition to get $$\psi_x(t):=\Psi(x,t)$$which presumably describes the full temporal behavior of the system at a fixed point $x$.
However a quick check shows that these functions do not form a Hilbert Space, since they are not square integrable. To see this, just consider a stationary state of the form $$\psi_x(t):= e^{-iEt}\phi(x)$$and integrate over time (remembering that $x$ is fixed) to get$$\parallel\psi_x(t)\parallel^2=\int_{\mathbb R}|e^{-iEt}\phi(x)|^2dt=|\phi(x)|^2\int_{\mathbb R}|e^{-iEt}|^2dt=\infty$$ In other words, when you try to interpret the solutions of Schrodinger's Equation as functions of time, in order to make the action of $i\hbar\partial_t $ meaningful, you don't get a Hilbert Space, which makes the entire subject of Hermitian operators and spectral theory meaningless.
