Trump
And if the Republican primaries had used ranked-choice voting, Trump wouldn't have won the nomination, since a majority of the electorate was against him, it's just that the anti-Trump vote was split.
Maybe. However, by the end of the primaries, Donald Trump was winning an absolute majority of the vote. Instant Runoff Voting (IRV) or plurality, an absolute majority is a winner. It's possible that ranked choice would have helped in the early primaries (and Trump would have never achieved the inevitability at the end), but it's by no means guaranteed.
If your goal is getting rid of Trump, then I would suggest getting rid of partisan primaries instead. That could have led to a candidate like John Kasich or Jim Webb winning.
IRV might have helped in the general election. Presumably Jill Stein voters preferred Gary Johnson and Hillary Clinton to Trump. And Clinton voters probably preferred Johnson and Stein to Trump. It's not clear how Johnson voters felt though. Libertarians have traditionally been split on second choice candidates. In any case, IRV is heavily dominated by the first choice. Since Trump and Clinton were the overwhelming first choices, it would have still been mainly between them.
Ranked choice in general
Ranked choice ballots are more complicated to fill out. Instead of filling out a single choice, voters have to fill out a list.
Ranked choice voting is harder to understand. It's not as obvious why one candidate wins over another.
Ranked choice is harder to process. Instead of just counting votes, the system has to count ordered lists.
Ranked choice doesn't fix problems with partisanship or oppression of minority groups in favor of majority opinion. But those are common complaints about the current system.
Note that these aren't insurmountable details. In particular, computers make the implementation difficulties much less important. But they are problems with every ranked choice method.
Also note that this doesn't include any of the arguments for ranked choice voting methods. This question only asks for arguments against. Balancing the competing arguments would be more opinion-based.
IRV in particular
Plurality is better than IRV by some criteria. In particular, monotonicity, consistency, and participation. Quick summary, there are tactical reasons to list different orders than one's actual preference.
IRV does not meet the Condorcet criterion. So in a high partisan environment, it tends to devolve into the same choices as plurality voting.
Of course, IRV is also better than plurality voting on some criteria. In fact, every voting system has some criterion on which another voting system is superior. In particular, the Condorcet criterion and Later No Harm are incompatible, so no voting system fulfills both.