I'm looking at portable CD players and am wondering - If I'm using "line out" to an amp to K701s - would the quality of the portable CD player affect the sound in any way? 
 
My understanding is that "line out" bypasses all the internals, and it's those internals which are usually crap in the cheaper players.  Since (if I understand correctly) I'm bypassing those components, would I be okay getting a cheap line out equipped player from Best Buy (and cheap 2 year replacement warranty) and going on my way?  
 
I'm looking at these two:
 
http://www.bestbuy.com/site/Insignia%26%23153%3B+-+Portable+CD+Player/8222915.p?id=1166840028182&skuId=8222915&st=cd%20player&cp=1&lp=5
 
http://www.bestbuy.com/site/Sony+-+Walkman+CD+Player/8234449.p?id=1168043378369&skuId=8234449&st=cd%20player&cp=1&lp=4
 
Thanks.

A line out doesn't bypass all the internals, it bypasses the analog section. This leaves the DAC as able to have an effect on the sound. Whilst the DAC would not normally have an appreciable effect, as the players involved here are very, very cheap it is likely that it might - a very cheap chip being used.
I would advise against creating "portable audiophile" setups around big, expensive headphones intended for home use. It is generally rather expensive and many "audiophile" DAPS (cough Hifiman) measure pretty abominably. Such setups based around CD players are almost unheard of.
It is entirely possible you are not intending to create a portable setup, and are simply going with portable CD players to save money. However, I don't think you're going to get a very good CD player for $20 (or $40 for that matter). It is entirely possible that it will sound great, but for $20 there are no promises on that front. On the other hand, it's $20 - if it sounds crap you can always buy something better.