Why should jet engine throttles be moved smoothly? On the only jet engined aircraft I'm trained on, the Bell 206B (Allison 250 engine), I was taught to move the throttle slowly and smoothly, especially when opening.  I've also see this done with Rolls Royce testing RB211s and early Trents.

I've realised I don't actually know why this is.  Is it to prevent possible flame out with a sudden increase in fuel with no corresponding increase in air mass since the engine will spool up much more slowly?

Do modern engines need this handling?  In this video at about 0:17, the pilot commences the roll by more or less slamming the throttle open. My guess is that like all other computer controlled controls, the FADEC interprets this as "as much fuel as possible without flaming out".
The issue is the possibility of a compressor stall.  The combustion will directly respond to changes in the thrust lever but it takes time for this to spool up the turbine that is connected to the compressor.  If the incoming air from the compressor is not of adequate pressure because it hasn't spooled up enough you risk a compressor surge.  In these engines you need to slowly increase thrust so that the turbine can spool up (and hence the compressor) so the airflow through the engine remains adequate for the engine power.

Modern engines with FADEC systems should not have these issues as the FADEC can enforce rate limits for fuel flow increases or directly calculate the maximum safe fuel flow from observed engine parameters.  Modern engines might also have more sophisticated compressors with variable stators that can help mitigate compressor stalls/surges.