I have mixed with 240DFs if you are going to _mix_ with headphones they are definitely the way to go. Mixing with headphones is not a good idea for all of the reasons that people always say it is not a good idea. But, I have done it with those headphones with good result. 240DFs sound flat, and pleasant. They are very unSony which I love. I have a bunch of sony cans for tracking but I have 240DFs for listening and eqing in them is a snap. They take a ton of juice and enaything that runs on a battery will be scraping big time to provide a sgnal without much bass. They run good off of the board of my Tascam m2600 MKII (an 8 buss analog mixer) and from plug in stereo gear.
I got into a better place now and I dont use them as much. I can be loud here. The place I go for information like this is headwize.com and something like headfi ? The guy that built the CMOY amplifier design is on one of those boards and he is good. For a very cheap amp the CMOY might be your bet but I have not heard _any_ headphone amps. So I will abstain from commenting. However, there is a slew of information and opinions for pairing headphones with headphone amps there and if you use the search function you will probably find an opinion what works best with 240DFs.