The five-syllable word is
This definitely relates to
Honey bees or other pollinators.
You might owe them an apology,
We think of bees as annoying, but their role is critical.
The saxophone can be a key,
The key of many saxophones is B-flat (thanks, @Arth)
I'm looking for a specific word,
Not sure here, perhaps Apoidea which is the taxonomic superfamily to which bees belong.
It has one of the world's smallest turds.
Maybe the "po" part of that word is like a really small "poo".
Bees are small things.
If you find one, leave it be,
Don't mess with the bee.
Feeding the world keeps it busy,
Bees pollinate many flowering plants that we depend on for food.
You can catch more flys with honey,
Many bees make honey.
Steal their own and make your money..
Totally happens all the time.
Five syllables, and one short word,
Apiology has 5 syllables and is relatively short.
Yes, my puzzle has a turd (ignore),
Ignored, as directed.
So timeless they have their own Latin root,
Apiology is from the Latin api = bee.
We study them, that we may have fruit.
Understanding bees and their behavior/diseases is crucial to getting fruit, since bees must pollinate most fruit trees/plants.
And the question's title:
Busy as a bee!