French does not seem to have influenced English pronunciation much (unless it turns out that French influence caused the Great Vowel Shift, as some scholars postulate), but this article will discuss what little we have found.
Without French influence it is likely that English would today substitute the initial /dʒ/ in loanwords for another sound. This is because /dʒ/ does not appear in the initial position natively in English. Evidence from Middle English indicates that some Englishmen may have substituted initial /dʒ/ with /tʃ/; jealous can be found in Middle English spelled as chelous.
/ʒ/ and /ɔɪ/
/ʒ/ (the zh sound) is not a sound inherited from Old English but a relatively recent addition resulting from French loanwords that historically had /zj/ such as measure and vision. The /zj/ sequence was never present in native words, so no native word has /ʒ/. The same goes for /ɔɪ/, which was introduced in Middle English through French loanwords, so words such as toil and destroy are clearly not native.
Latin G vs French G
Words with ⟨g⟩ borrowed from Latin, like Germania, should be pronounced with /g/ or perhaps sometimes /j/, but not French /dʒ/.
Old English Z
There is some evidence that ⟨z⟩ in Old English, though very rarely used, was pronounced as /ts/. Perhaps this is how we would pronounce ⟨z⟩ in loanwords today.
French Influence on Stress
Supposedly, some words like mankind had their stress changed due to French. Here is a relevant passage from a book: "It is therefore often said that Middle English replaced the [Germanic Stress Rule] with a Romance Stress Rule (e.g. Lass 1992: 83-90), which it imported together with a large number of French and Latin loans in the wake of the Norman Conquest."
French ⟨u⟩ vs English ⟨i⟩
Sometimes Old English ⟨y⟩ became ⟨u⟩ instead of ⟨i⟩ under French influence, and this sometimes affected a word's pronunciation.