What to do when your opponent doesn't play along in the opening? I looked around and could not find an answer for this.

I've studied a pretty good amount of opening theory and so I know the "correct" moves for some basic lines.  Most e4 openings and a fair amount of Nimzo-Indian, Pirc, and QGD for D4.

The thing of it is, I often play opponents who have not studied these lines.  So they play the "wrong" moves and I'm not sure how to proceed.

Is it best for me to stop and eat clock time figuring out why the wrong move is wrong?  Or should I keep playing my same opening even though I might be putting myself deeper into a hole?
It sounds like you're a beginner. As such, you should be focusing more on tactics than openings.

First, just because they play a move that's not in your book, doesn't mean it's "wrong". It may be an older line that's not in modern opening books, or just not in the books you have.

That being said, to answer your question, it depends on the opening. In a slow, closed or positional openings, it's not likely to matter too much, when your opponent deviates from your book, just adjust your plans as necessary. If it's a gambit, or tactical opening, you should take some time to check to see if it's a blunder.