[grzm]

In the dialect I speak (roughly North Central American), syringe has the stress on the second syllable, which makes the rhyme with orange (ˈôrənj, ˈärənj) a bit awkward. Looking it up, I was surprised that both stress on the first syllable and on the second were listed.

  səˈrinj, ˈsirinj 

It piqued my curiosity. May I ask where you've heard the latter?

[howlgarnish]

Interesting. Now that I think about it, /səˈrinj/ sounds quite American to me, while /ˈsirinj/ is more English?

Also, you characterize the 2nd vowel in "orange" as a schwa, but to many people it's /i/:

https://en.wiktionary.org/wiki/orange#Pronunciation

[grzm]

I just lazily copied and pasted the phonetics from the New Oxford American Dictionary :)

[naniwaduni]

It's generally in the range of /ɪ/~/ɨ/. Descriptions of English regularly conflate the ɪɛʌ with ə as they behave somewhat similarly wrt stress-related sound changes.

[thaumasiotes]

I agree with the issue of stress, but it seems less problematic to me than the vowel mismatch. Orange and door use the FORCE vowel in their first syllables. Syringe doesn't; it uses NURSE, and this is enough to prevent the words from rhyming.

(Perhaps you view orange as using the START vowel in its first syllable. That's still not NURSE and still doesn't rhyme with syringe.)