This answer is based on the events shown at ~00:00:21 in The Simpsons opening sequence for the 1990-91 season which was largely the same from 1990-2009. Other season opening sequences may vary¹. In all seasons, the opening sequence introduces the characters by showing a brief microcosm of their personality traits. e.g. Homer's aversion to work, Marge's motherly instincts, Bart's detention, Lisa's extra-curricular activities, etc.
In the early seasons¹ of The Simpsons, Montgomery Burns does a double-take on his watch when the Springfield Nuclear Plant's shift whistle blows in the opening montage.
A factory or plant 'shift whistle' is synchronized with a central time clock. These often have an integrated in/out time card punch system associated with the drudge of boring, menial work. While The Simpsons is a cartoon, it really isn't styled as SciFi or future tech so it is highly unlikely that Mr. Burns' watch controls the shift whistle.
Mr. Burns apparently doesn't think it is quite time for shift end. Worried that he is not getting a full working day from his employees, he looks at his watch. Concerned that his watch is not showing the correct time, he holds it to his ear to see if it is ticking and then shakes it as if that it somehow going to instantly correct the situation.
I see at least three possible explanations.
The watch is showing the correct time; i.e. the time that the current shift of workers is over. However, Burns' megalomania isn't going to allow him to admit to himself that he's wrong in his own estimation of the time of day so he blames it on his watch. There is no direct correlation between his watch and the plant's time clock; they are simply both set to the correct time.
While this goes to show Burns' arrogance, I don't think it is the most likely explanation of events.
His watch is either slow or stopped and this led him to believe that it wasn't time yet for that shift of workers to go home. He puts it to his ear to see if it is ticking and then shakes it as if that was going to instantly correct the displayed time.
I find this to be the most likely explanation and it shows Burns' impatience with anything he feels does not live up to his standards. His attitude towards his non-functioning watch is analogous to his attitude toward his workers.
Burns' self-winding watch appears to have stopped. He puts it to his ear to confirm this then shakes his wrist with the watch to get it going again.
I find this the least likely scenario. It relies on the viewer being familiar with self-winding watches and doesn't offer much in terms of Burns' character.
¹ The scene with Montgomery Burns wasn't included until the 1990-91 season. In the previous 1989-90 premiere season, there was a co-worker eating a sandwich with the help of some forge tongs. For 2010 and later, some background scenes became transient and Burns and Smithers were replaced with other scenes; one involving Lenny and Carl with the 'X Days Without Accident' sign.