Omoide Yokocho in Shinjuku has become a bit touristic, but still able to experience atmosphere of local Izakaya. The good point here is that because there are lots of tourists, people are used in foreigners.
Tsukishima is known for Monjayaki, but there are small Izakayas in narrow streets if you turn directions from the main street, Monja Street.
Tsukiji area might be quiet comparing to Shinbashi in terms of Izakaya.
Any Izakaya in Shinbashi is authentic, Shinbashi is the mecca of Izakaya in Tokyo. Lots of businessman gather in the area esp. on Fridays.
Walk along the railway from Shinbashi to Yurakucho. You can find lots of small taverns filled with people. That is the most easiest way to find Izakaya with authentic atmosphere.
Another area in Shinbashi is going from Shinbashi station to direction of Toranomon. The district is filled with small and big Izakayas!