After Oliver canceled their dance lessons, it's clear there's conflict brewing between Oliver and Shane that only addressing its source head-on is going to resolve. From the opening moments of The Edge of Forever, it's clear they're on a crash course for that confrontation. But before they come to blows, we do get a sense here and there of how each of them is attempting to process the situation.
For instance, Oliver considers the cancellation an "unexpected development" and a "sensitive situation that requires a certain sensitivity," a sensitivity which Shane is hard-pressed to be conscious of given her own feelings on the situation.
One of the things I found brilliant about this scene is how it starts with Shane pulling out her tablet and looking Holly up. Remember Shane's House Rule was to "get [her] head out of the computer and dance a little," a directive to leave her comfortable tech shield behind and test the waters of human connection. After boldly accepting that challenge, it seems to have backfired, Shane putting up her shield once again as the scene begins to unfold. It's also the action which begins to draw ire from Oliver, setting the argument in motion to some degree. Notice it's the same person who caused her to withdraw that is calling Shane out on withdrawing.
When it comes to mixed signals, Oliver is the accidental master. Remember that the day of the showcase Oliver cites the lessons being a gift from Holly and "accomplishing" what he set out to do as the reason for canceling. By Monday, however, a more nuanced narrative emerges that causes more questions than answers.
Think of it this way---when Oliver began taking dance classes with his previous partner, he had every intention of participating in the showcase. He wasn't concerned with "appearances" or "sending the wrong message." This concern emerged only after Shane was involved. If Oliver inviting Shane to be his partner were purely functional, "appearances sending the wrong message" would have never entered the discussion. Along the same lines, Oliver labels his decision to make Shane his partner “ill advised” on his part, suggesting his head wasn’t the only part of himself used to make the decision---*cough* his heart *cough*---whether he can acknowledge it or not. We'll talk more about this soon, as it ties into One In A Million in a very particular way, but for now know that this is only the first time Oliver's heart trumps his head, only later causing him to backtrack.
My Quarrel With You Is...
Believe it or not, Shane and Oliver are mad at both Holly and each other throughout the course of this interaction.
Shane begins angry with Holly for having signed the lease, then Oliver for remaining in a visible state of denial, a painful reminder of wounds inflicted in her childhood and knowing what it's like to "wait for someone who never comes." The fact of the matter is Shane is hurt both for him and for herself, and her actions and words to Oliver reflect that. She wants him to be able to move forward, which he seems unwilling---but is really afraid---to do.
As for Oliver's response to Shane, it's clear he has placed an expectation on Shane and her behavior that this incident violates, deeply upsetting Oliver to the point of verbally alienating her. He attacks Shane's computer and her "insatiable curiosity," which he previously labeled in the Pilot as an asset. Oliver expressly states to Shane in From Paris With Love that breaking her promise to him was the source of his issue with her there, citing her "curiosity" once more. In both The Edge Of Forever and From Paris With Love, Oliver seems to take direct issue with Shane's actions, fueled by their consequences for him.
Insult To Injury
A troubling pattern of behavior to emerge in this argument is Oliver's tendency to add insult to injury. We know that one thing Shane craves is knowing where she stands in her relationships with others. After being abandoned by her father and remaining in limbo in her relationship with Steve (which we only learn later) it's obvious how this need developed. For Oliver to not only call their dance partnership a "mistake" but to later suggest even their friendship was in jeopardy---if they were even ever friends at all in Oliver's mind---was devastating and just plain cruel.
It may have simply been an argument to us at the time, but Shane and Oliver's confrontation in The Edge of Forever sets precedent for future arguments, points of tension in their relationship and how they are addressed, which we have seen play out as the series progressed.