The problem will occur if ##H## is dependent on time. We must perform derivative ##H## in the component ##\exp\left[-iH(t-t_0)\right]##
$$ i\frac{\partial}{\partial t}\exp\left[-iH(t-t_0)\right] = \frac{\partial H}{\partial t}(t-t_0)\exp\left[-iH(t-t_0)\right]+H\exp\left[-iH(t-t_0)\right] $$...