The question is, is there any time he SHOULDN'T shut up (the answer is also no)

If I was in a room with bin laden, hitler, and wesley crusher with only two bullets, Id shoot wesley twice. -coldgoron -michael scott

SHUT UP WESLEY!

Wesley looked around as he heard a deep and drawn out rumble….”shuuuuuuuut uuuuuup”. He gazed upward finding the imposingly disgruntled expression of none other than Treebeard, Shepard of the Trees.

If a Kolvoord Starburst goes off in The Neutral Zone
and there's no one there to see it,

Must Wesley still shut up?

Yes

He would still have to hear himself. Truly, the worst fate.