I myself am a fan of the Red Eye as well... I wouldn't say I use it over mic'ing, its just an excellent tool for blending and for fixing problems with others' work. I use it, as I said earlier (I think), to amp acoustic parts later if a blend of the DI and the mic isn't doin it for me. I use it on occassion with guitar, and it is nice to have that extra channel of completely unaltered playing, in case I want to change the tone of a guitar part later (without having to rerecord the part), but this is usually rare.
I've also used it when I recorded a guitar part that was clean, but the amp was producing a lot of hum that day... I have the odd situation where my amps normally sound perfectly fine in the house (split-floor apartments), but on occassion are extremely noisy (must be somethin the guy above or below me only runs infrequently). So, in that situation I will DI and record the amp too (just to get a feel for the tone), and then reamp later when everything is running hum-free.