I use Amplitube with M50 headphones and was always half-dissapointed with the results... it's just missing that "real" feeling, almost like a lack of mids.

Today I was playing through my monitors (Mackie CR3) and noticed that it was tremendously closer to that real tube feeling, probably due to the fact they were bigger drivers and the CR3s do have a little bit more low end than a flat response imo.

How crazy of an idea is it to try to mic up my monitors rather than recording directly into a DAW? I don't have a mic, but it could be something worthwhile. Or at least an experiment.

Just a thought.
Anything but crazy. I've done it with synths and other sounds to give them more 'air'. If I'm not mistaken Quincy Jones (producer of Micheal Jackson a.o.) did it with all his synths.

You just try it out! Might experiment a bit with the positioning and pre and post-EQ'ing but if you layer it with your DI-Amplitube sound it can get you some pretty good results.

There is absolutely nothing wrong in miking up a monitor with a decent mic. If you can get the kind of tone you want, then that is what you should aim to record!
It would be great if you could have a pair of headphones with a flat frequency response to listen to what you have recorded. I listen to all my mixes through my headphones just to make sure that my room is not colouring the sound that I am playing back through my monitors
You might need to make sure that the mic is not routed to your monitors while recording just so as to avoid feedback.