There seems to be a longstanding belief that, if a pitcher has warmed up and/or begun to pitch, but then there is a long rain delay, that he should be replaced after the delay instead of continuing to pitch. The concern is that pitching after he has "cooled off" may be ineffective or risk injury. (A related question, unanswered so far: What has historically been the effect of a long rain delay on a pitcher's arm?)
How does this interact with the new three-batter minimum rule? If a pitcher enters the game and there is a rain delay before he has pitched to three batters, is he still required to continue pitching after the delay?
One could ask the same question about suspended games that are resumed on a later day.
The three-batter minimum rule (5.10g) does not appear to contain any exception for delays in the game, but maybe there is something I've missed. I could also imagine an unwritten rule that the umpire would waive the three-batter minimum, or that a cooled-off arm could count as an "injury or illness which incapacitates him from pitching".
This almost(?) happened in the Giants-Reds game that is going on right now (July 17, 2023). The Reds manager went to the mound to make a pitching change, and just then, the umpires called for the tarp. On the other hand, I think the umpires may have called the delay before the name of the new pitcher was announced. If so, presumably when the delay ends, the Reds can insert any available pitcher - not necessarily the one who was warmed up before the delay. I wonder if the umpires did this deliberately as a way to avoid the issue?