Hey, I have an AKG 240M.
I'm glad to find someone still using them.
About your question, since I also have the DV332, I can tell you that, IMO, this pair is really a good combination.
But frankly, I don't think you really have to get DV332 to make your 240 sound good since mine sounds reasonably good from my CDP phone-out and my NAD C372 integrated amp (which drives my HD650 horribly). I think Figaro or other lower model should be sufficient. However, if you decide to get 332, I can assure you that your cans won't sound bad with this particular amp.
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I've just plugged my cans in to listen carefully again to tell you what they actually sound.
Well, Vocal is warm, sweet and lively, making it very enjoyable experience listening to Audiophile-vocal albums.
In addition to a good vocal, pairing with DV332 could also bring out more of those "sparkling" treble and deep bass. Of course, bass is not as deep as those produced by my HD650, but I can say it's very satisfactory bass.
One last thing, it seems DV332 makes this cans sound "faster" and smoother. Another benefit is that I have to crank the volume up to 11-12 region to get a decent volume (for me, which is a bit loud). By turning it farther, I get more defined control.
I also compared my DV332 with my CDP phone-out and NAD C372, and I have to take back my words, as both of them are significantly inferior to DV332 when driving K240.
FYI, I'm using Svetlana 6S19P-V as power tubes and LM Ericsson 403B as preamp tubes, but I think you sould get the same result using other good tubes like Ulyanovsk 6S19P and WE 403A/B.
Some picture
Hope this helps you decide.