Pronounced alone (with no surrounding words), the American English "for" has a deep "r" at the end. But what about

for free? 

Can the "r" in "for" be omitted in this case ( [fefree] )? Or is there a shade of "r" still left?

The dreaded Rs. I have never been able to explain what I do to another. When the vowel is u-ish, my tongue seems to move back and block the airway, almost into a glottal. 

 

I probably do discrimate between 'four' and 'for', though probably as an artifact of spelling. 'For' is smooth, with some nice lip rounding. 'Four' is a diphthong, with the u realized with the constrictive sensation at the back of the mouth, and with the u left almost unrealized. I can do a smooth 'four', but a 'w' is inserted, and its essentially a homophone of 'foe-were'.

 

'Fir' and 'fur' have subtly different vowels. 'Fir' is higher and flows smooth. 'Fur' has the tongue pulling back against the airflow; if I exaggerate, I can feel the throat muscles constricting.

 

To answer Juozas' question, in rapid speech, I can reduce 'for free' to /fəˈfri/.