We report on infrared spectroscopy experiments on the electronic response in (Sr1-x La x )2IrO4 (x = 0, 0.021, and 0.067). Our data show that electron doping induced by La substitution leads to an insulator-to-metal transition. The evolution of the electronic structure across the transition reveals the robustness of the strong electronic correlations against the electron doping. The conductivity data of the metallic compound show the signature of the pseudogap that bears close similarity to the analogous studies of the pseudogap in the underdoped cuprates. While the low energy conductivity of the metallic compound is barely frequency dependent, the formation of the pseudogap is revealed by the gradual suppression of the featureless conductivity below a threshold frequency of about 17 meV. The threshold structure develops below about 100 K which is in the vicinity of the onset of the short-range antiferromagnetic order. Our results demonstrate that the electronic correlations play a crucial role in the anomalous charge dynamics in the (Sr1-x La x )2IrO4 system.