I have an HTC One. Based on your findings, it should run the Audioengine D3 fine?
Specs say 200 mA current draw: http://audioengineusa.com/Store/D3-24-Bit-DAC
Only concern I have is that my HTC One doesn't work with my Schiit Modi, which according to their website states only 120 mA current draw. Also, I've tried using the Aqvox USB power supply and it still doesn't work. Maybe it doesn't recognize the external power, or maybe besides power, there are other specifications also needed for OTG to work?
http://schiit.com/products/modi
http://www.aqvox.de/usb-power_en.html
Any device that draws less than 300mA should work fine with the HTC One regarding the power requirements.
Besides the power requirements, Android's implementations of kernel-level or OS-level features can affect the compatibility of different USB DACs.
Android 4.1, for example, handles USB audio differently than Android 4.2 (I do not know the cause, but some DACs might work with Android 4.2 when it won't work with 4.1 on the same phone (also, this could be due to kernel reasons, though)).
Many devices are lacking USB Audio support at the kernel-level such as the 2012 and 2013 Nexus 7. When I loaded CM10.2 on the 2013 Nexus 7, USB Audio support was included in the kernel that came with the ROM. To enable USB Audio support, the 2012 Nexus 7 had to use a kernel that was modified to include USB Audio, such as some of Morfic's kernels and Timur's kernels.
Unfortunately, Android USB Audio is not straight-forward. If you have a DAC that won't work with your phone, try flashing CM10.2 or similar ROM for the phone. Chances are, USB Audio will work with a particular kernel/ROM combination.
Asus HD7 Tablet with latest 4.2.2 Android does not recognize Fio E17.
Verizon Note II with Pacman nightly ROM recognizes the Fio E17 and stops playing through the Note II speaker but the E17 has no headphone output using the stock Apollo music player.
I can only comment on the Galaxy Note II regarding the E17. The Note II can fully power the E17. I have found that the E17 works with UARP on the Note II. If using Android 4.1.2 on the Note II, the E17 will not work. I think the chances are high that the Note II will work with the E17 if using a different ROM/kernel combination. I would first try a stock-based 4.3 ROM, then move to CM10.2 if that doesn't work.
I should add that the DAC in the Woo Audio WA7 works with my HTC One and I asked them the current draw and it's rated at 200 mA.
I tested the Fiio E17 with the HTC One. I found that the current draw was 130mA with USB charging turned off on the E17 and 220mA with USB charging turned on. The HRT Music Streamer II+ will not work with the HTC One, and it draws 370mA.