No, there is no requirement for log entries to be idempotent if you are using a distributed consensus algorithm such as Paxos or Raft. There are many advantages to designing your system around idempotent operations, but it is not always possible to do so. Distributed consensus algorithms are a great fit for cases where you cannot achieve idempotence so that you need to be sure that operations are processed at most once on each replica. Moreover they let you ensure that the operations are always processed in exactly the same order on each replica. These are very strong properties, and you need to perform some relatively-expensive coordination to ensure that they hold, which is why we try and avoid them wherever possible.
Distributed consensus guarantees that each log entry will be the same on every replica as long as that log entry has been accepted by a majority of the replicas. Each replica must only process log entries that have been accepted by a majority of replicas, because other log entries might change during a recovery. Each replica must carefully keep track of which operations it has already processed to avoid processing them again during a recovery. This is simple to do since the log is totally ordered, so each replica can track the operations it has processed with a single number representing the position of the last processed operation in the log.
In fact, another way of looking at distributed consensus is that it is an effective way to add idempotence (and commutativity) back into a collection of operations that might not be idempotent (or commutative). So no, the operations don't need to be idempotent because you can obtain the idempotence you need from the distributed consensus algorithm instead.