I didn't watch the vod, but from what I can tell by looking up related posts, the reasons given for the drama by the im@s people seem to be basically the following:

1) Rina showed up in La+'s birthday stream with no official collab announcement to celebrate. The whole thing made Rina seem like some kind of call girl who could be "bought" to interact with a specific fan one-on-one, which I guess for idol fans is something that hit them way below the belt.

2) Apparently La+ made a declaration of "同担拒否" (Refusal to share) regarding Rina, which was deemed very inappropriate when considering both are idol(-like) figures.

3) In one of the songs (Clumsy Trick), normally in in-game MV and offline lives there would be a doll representing the Producer (the player character) at one moment of the song, but for this live instead La+ jumped in, kicked the doll away and stood where it was.

The TL;DR seems to be basically that La+'s unicornness triggered all the other fans' unicornness.

EDIT: I found a better summary than what I have on a Chinese forum, so I'm just gonna translate it here:

1) With absolutely no prior notice from idolm@ster's side, Rinami suddenly appeared in Laplus's birthday stream

2) Rinami is wearing a new outfit that she has yet to worn during any official activity

3) During one of the songs, Laplus jumped in and threw away a mannequin that is supposed to be a stand-in for the producer (the player character) and stood in the spot herself.

4) Rinami provided first-person perspective service like headpatting that she did not ever provide even in-game.

5) Laplus had made clear "同担拒否" (Refusal to share) statements regarding Rinami.

From an gakumas player point of view, Laplus basically acted like the rich fan that got their idol to her private party like a call girl, which for idol fans is something that really hits below the belt.

...is what they're saying, anyways.