I always assumed it was a sign that an otherwise unassuming and placid teacher had a ruthless streak and could deal swiftly and, if necessary, brutally with a situation.
---The book explains it (and it was written pre-Harry Potter).
The owl had been nesting in the disused chimney.
When Prideaux fired it up a half-dead owl was smoked out onto the classroom floor, shocking and startling the boys.
Prideaux whisks it up and out of the room.
Later the boys find the dead owl on the rubbish heap, its neck neatly twisted and broken.
The point is that Prideaux is a skilled killer.
Also in the book, Haydon is killed by persons unknown, his neck neatly twisted and broken.
The movie version compresses the more nuanced details from the book (that also foreshadow the different details of Haydon's death).
But the essence that Prideaux is a capable and ruthless killer when required are preserved in a simpler, shorter scene.
movies.stackexchange.com/users/134/matt_black
The significance of the bee scene is that, unlike the others in the car, Smiley is totally focused on the things that matter (the mole) and able to completely ignore trivial distractions like the bee.
movies.stackexchange.com/users/624/Simon Whitaker
In a flashback we see Jim is tortured wearing the headphones while being exposed to disturbing screeching sounds.
Something snaps when he hears the owl making a similar sound.
movies.stackexchange.com/users/40918/Lisa Douglas
The owl scene is much more subtle, it is introduced right at the beginning of the book.
At the end, it is never explicitly revealed who killed Bill Haydon
but it is strongly implied "Stooping over him, he caught the smell of alcohol -
he guessed gin or vodka -
(Prideaux drank vodka)
...
It wasn't till he tried to lift him that his head flopped over, and the rest of him followed as dead weight"
(Prideaux had wrung the owl's neck).
QED
it must've been Prideaux wot dun it.
movies.stackexchange.com/users/5054/mantonman
I say the owl, disturbed from his chimney, is a metaphor for betrayal or a secret let loose, a symbol of wisdom and a portent for Jim Prideaux of exposure, hence his prompt execution of the bird.
movies.stackexchange.com/users/28786/Katy Burton
There's a scene in "Tinker Tailor Soldier Spy" where Prideaux (Mark Strong) is in the class room teaching, then suddenly an owl pops out of the fireplace. Prideaux quickly dispatches it. Later on, in the same class room, Prideaux is talking to Smiley (Gary Oldman) and we see a stuffed owl on the wall.  That (and the bee scene) always seemed rather random to me. Or am I missing some deeper meaning?
Significance of the Owl in "Tinker Tailor Soldier Spy"
movies.stackexchange.com
<plot-explanation><analysis><tinker-tailor-soldier-spy>
movies.stackexchange.com/questions/896
movies.stackexchange.com/users/29/System Down
 I say the owl, disturbed from his chimney, is a metaphor for betrayal or a secret let loose, a symbol of wisdom and a portent for Jim Prideaux of exposure, hence his prompt execution of the bird. The owl is a symbol of wisdom and a portent of exposure
The owl is a symbol of wisdom and a portent of exposure for Jim Prideaux, a metaphor for betrayal or a secret let loose.