We explore the photo-production of possible exotic $N^*$, which are narrow pentaquark candidates containing light quarks or Charm or Beauty. The latter two are also known as $P_c$ and $P_b$ states in literatures. We analyzed the Compton scattering off the proton in the third resonance region in a coupled-channel effective Lagrangian model with K-matrix approximation. The evidence of exotic $N^*$ with light quarks are found to be weak at present. Motivated by the $P_c$ observed by LHCb, we discussed the possible signal of $P_c$ and $P_b$ states in $\gamma p \to J/\psi p$ and $\gamma p \to \Upsilon p$, respectively at an Electron-Ion Collider (EIC) in order to disentangle the nature of these states.