I remember this song very accurately when I was younger...that does't mean the version I learned was unchanged or more correct, but there is one part that I am pretty sure is the most correct:

"Liniment" is a medicinal liquid, theoretically clean and not "linty" (albeit these ARE bugs), "lint" is the old way of saying the cotton padding (originally from a byproduct when refining cotton) used in dressing wounds. So "liniment and lint" should be the correct version.

One other thing I remember for sure, the version I learned used "smashed", not "fell", I saw both there. I remember learning the verb then and thinking it was really cool.

One thing I'm not sure about is a version I haven't seen above is, when the bugs are shouting, whether they should take the aspirin was a question that was asked, which is, according to the teacher's explanation, what spurred the bugs to call a doctor in.

So, this is the version I am pretty mostly sure I learned in elementary school, 20+ years ago.

Mr Fly climbed up a tree,
climbed as high as high could be,
Lost his grip, came tumbling down,
Smashed to pieces on the ground.

When the insects heard the sound,
Echoing for miles around,
They began to buzz and cry,
"Quick! First aid for Mr. Fly!"

"Get a bandage, get a splint!"
"Get some liniment and lint!"
"Should we give him aspirin?"
"Someone call the doctor in!"

Then a wise old flea spoke out,
"You don't know what you're about.
He's beyond the reach of aid,
Get a pick and get a spade."

Then the insects finally knew,
What they really had to do.
On his tombstone bears the scrawl,
"He who climbs too high must fall."