Why do some anime not have an opening in the first episode? A lot of anime I've seen don't have openings in the first episode, instead the openings start from the second episode. I cannot particularly name any of them besides Boku Dake ga Inai Machi.

Why do some anime not have openings in the first episode?
Several reasons: Production cost, first impression, spoilers.

Theme songs are commonly used to save money on production. Draw that opening once and show 90 seconds of a roughly 22-minute episode every time. And there is also an ending which is a double value :p

When a series is only starting, you might want to make it interesting right away, and not just show the opening that the devoted viewers are already used to, and won't switch the channel if they're not entertained enough by it.

If the opening animated sequence features content that might need at least some introduction and explanation, that is planned to happen in the first episode, it would be good to use those additional 90 seconds for a more natural development.