Is there any emergency situation in which jet engines would not be shut down before passenger evacuation of a transport aircraft? A comment to this answer discusses the possibility of evacuating passengers risking ingestion into the engines if evacuating over the leading edge of the wing. That particular question discusses the Boeing 737. For example:


  "I always thought that the reason to go off the back of the wing was to avoid getting sucked into an engine."


and


  "I'd imagine on a land landing, the leading edge could bring a passenger dangerously close to being ingested, should the engines be rotating.."


However, I really can't see any valid situation that calls for the following sequence of events:


Emergency landing (either on land or on water)
Keep engines running
Begin evacuation of aircraft
Shut off engines (during or after evacuation)


Instead, the sequence of events would seem to be:


Emergency landing (either on land or on water)
Shut off engines
Begin evacuation of aircraft


in which case some unlucky individual might possibly hit (come into contact with) some external or exposed part of the engine or engine pylon, but I have a really hard time seeing how anything could be ingested into the engine at that point.

For the purposes of this question, I suppose "engine running" can be interpreted as "sufficient intake airflow for large foreign object ingestion to be a concern".

Is there any valid situation where post-emergency-landing evacuation would begin while the engines are still running? If there is, then please describe at least one situation which would call for that and describe why such a course of action would be necessary in that situation as opposed to waiting for the engine front intake fans to spin down.
It is possible for the engine to not respond to shutdown sequence, so you'd have to get the passengers off before dealing with the rogue engine. 

In the Qantas Flight 32 one engine had an un-contained turbine failure in engine No. 2 leaving them unable to shutdown engine No. 1. This was due to a piece of debris severing all communication to the engine, essentially leaving it out of control and running.

This in turn made it impossible for the crew to activate any safety mechanisms for the engine. It was eventually shutdown by dousing it until flameout. 

Report by Australian Transport Safety Bureau

Wikipedia Article on Qantas Flight 32

It should be noted that in this specific case they didn't evacuate immediately due to a fuel leakage and the overheated brakes, though when they did evacuate engine No. 1 was still running. 

Another situation where immediate evacuation may be preferable is if the aircraft had a cabin fire, meaning the highest priority would be cabin evacuation, which could potentially happen simultaneously to the crew attempting engine shutdown.