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?
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.
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.)
The rhyme has to be from the last accented syllable (otherwise "tinge of fringe in your minge" are rhymes), so syringe isn't, and door hinge only works if you make it a compound word and make hinge unaccented, which means dropping the h. At that point you're in near rhyme territory.
> door hinge only works if you make it a compound word and make hinge unaccented
Rhymes are often judged in the context of spoken language, under which conditions, dropping the h and having hinge be unaccented relative to the preceding door wouldn't be unreasonable or uncommon.
