Right now, I own a Sony STR-DH520 (I haven't unboxed it yet). The impedance rating for the headphone output is not specified.
http://store.sony.com/sony-str-dh520-3d-surround-sound-av-receiver-zid27-STRDH520/cat-27-catid-EOL-Home-Theater-Speakers-Components
The other receivers I'm looking at are the Yamaha RX-V773WA
http://usa.yamaha.com/products/audio-visual/av-receivers-amps/rx/rx-v773_black_u/
and the Marantz SR5005.
http://us.marantz.com/us/Products/Pages/ProductDetails.aspx?CatId=AVReceivers&ProductId=SR5005
In this thread, Yamaha receivers have mixed recommendation in terms of audio quality. The 773 has Silent Cinema. No impedance rating for the HP output on the 773 or the Marantz SR5005. All three receivers say the headphones can operate in direct mode; I'm going to assume they also mean that the headphones are powered by the same amps as the other speakers.
For speaker amps, I'm looking at an Emotiva UPA-200.
My budget for headphones is in the $100 entry level range. I am going to be using Polk 6750s for music, movies, and most gaming, and closed circumaurals when I am streaming game play to avoid sound leakage and feedback spikes from open sets. Since the application will be limited to games and not music, I'm looking for a solution that has above average bass prominence, and overall neutrality otherwise. I'm also looking for higher power in the phones than average, since there will be less subtlety (and honestly, fidelity is a low priority for games in my experience). The genre of games I play are generally warfare titles, although there are games I play that do feature quieter audio, but aren't ones I'd stream (e.g. MO:UL), so I'd be listening on the 6750s. My budget for headphone amps if I actually need an amp outside of an HTR hasn't been set yet--I'm trying to get an idea of what it will take to get decent audio that meets the above quality guidelines.
Half of the headphones recommended in entry-level lists recommend headphone amps. One such set in consideration is the Fostex
T50RP, for at least $104.88. As far as headphone modding, let's say I've been less than successful in that arena. The closest I've ever come to repairing a headset was resoldering a mic while leaving the headphones nonoperational. My hobbies are more in the software and content creation side, so I'd rather spend $150 on a set of headphones that are ready for use rather than $100 on a set that need another $50 to improve their quality, even if it means the $150 pair are inferior to the $100 with $50 of modifications. I'm trying to shop with a whole budget in mind, and get an idea for a minimum budget that will serve the application decently.
EDIT: A day after I posted this, I tore the wires out of my KOSS supra-aurals. An hour and a half later, they're back to their normal rattling selves.