I am English. We absolutely do pronounce the r of her. Thinking about it, the sound is I think sometimes slightly lighter/shorter in the word hair (as in on your head, etc). By the way, as far as I am aware there is never an r sound with Je. There is a link above for forvo but perhaps a moderator could give an easy phonetic.
The discussion started when @Littlestrike was comparing the vowel — and only the vowel — in the English word "her" with the vowel of the French word "je", which are quite similar. The -r was not in the discussion until @tu.8zPhLD72zzoZN mentioned it, possibly because he misunderstood what the original poster (OP) wanted to say. The -r is irrelevant to this discussion as @AmbassadorTigger said.
What I said just tries to give support to what OP tried to say, but maybe it failed.
Anyway, you're right, there's no -r in "je".
About the phonetic part, those words are expressed phonetically as follows:
her — UK stressed /ˈhɜː(ɹ)/ ; UK unstressed /ə(ɹ)/
je — /ʒə/
As you can see, the vowel is the same when unstressed.