Does handlebar tape go inside the bar, under the bar-end shifter, when using bar-end shifters?
Most bar-end shifters require friction between the bar's interior surface to stay put, so no. Install the shifters securely and tape the shifter housing about where you would like it to emerge from the bar tape. You will end up wrapping over the first few inches of shifter housing before it loops forward and may require a bit of futzing to get it looking right.

Here is a lovely video showing shifters being installed on a drop bar.