The Hamiltonian operator in quantum field theory is constructed by using the fields in question. For example, the Hamiltonian for a free scalar field reads \begin{equation} \notag H = \frac{1}{2} \int_V d^3x \left( \pi^2 + ( \partial_i \phi )^2 + m^2 \phi^2 \right) \, . \end{equation} The key observation is now that a given field evaluated at different times does not commute with itself $$ [\phi(t',\vec x) , \phi (t,\vec x)] \neq 0 \, . $$ Therefore, the Hamiltonian evaluated at two different instants of time does not commute: $$[H_i(t) , H_i(t')] \neq 0 . $$
(The field commutator only vanishes for spacelike separations but not for timelike separations. See, for example, page 37 in Tong's notes.)