Short answer: no.
Long answer: Shielding works by surrounding the internal wiring of the guitar in a conductive surface that is shorted to ground. Stray electromagnetic interference (EMI) that would normally have passed through this "Faraday cage" is grounded by the shielding and thus does not pass through it into the circuitry that the shielding encases. The Tremolo system is likewise grounded, and as such isn't part of the circuit - no signal that passes into the tremolo or the strings will pass into the circuit because it is grounded, so that whatever emi might be present in the strings is drained to ground also. For that reason, shielding the Tremolo cavity will be a pointless thing to do - as the purpose of shielding is to protect the signal that has been induced in the pickups (and subsequent circuitry beneath the pick guard) - that signal does not pass to ground, but is returned to the amp.
So, shielding the tremolo cavity doesn't do anything, because there isn't any signal there to protect.
A note about shielding: it protects only the
circuit under the pick guard from the unwanted EMI. The lion's share of your EMI is going to enter into the circuit through the pickups themselves. In other words, shielding will reduce, but not eliminate EMI.
A better option for noise reduction is a hum-cancelling circuit - either by way of humbucking pickups, or by pairing alternately wound pickups together in the same circuit. (reverse wind coupled with regular wind = hum-cancelling). Many guitars do this already.
EMI is most noticeable when your volume is up, but you're not playing anything on the strings - that 60Hz hum (HUMMMMMMMM) - seems obnoxiously loud. Yet when you're playing something, that "hum" contributes to what some would describe as an "airy" tone. Take it away, and the resulting (clean) tone may seem to be missing something. It may sound muted, or lacking a sort of "live" flavor. It isn't a subtle difference either. You notice it. When your tubes start to crack, that same EMI that sounds so awful when you're not playing anything suddenly gives you're signal a difficult to quantify "live" sound that is diminished in guitars that effectively remove the EMI.
Many guitarists who have come close to eliminating the EMI, then back off because their tone sounds comparatively dull without it.
I expect that the reason this isn't done at the factory is because it is neither cost effective nor overly effective as an EMI solution. It certainly helps - don't get me wrong - but not as much as a hum-cancelling pickups (z-coils for instance) or pairing regular and reverse wound single coil pickups via a pickup selector.
That's how I see it at least.
(edit: I repeated myself on the tone stuff because I left the post half written to eat lunch, and came back to finish it and ended up repeating myself - so I blended what I said before and what I added after into one paragraph).
Last edited by DanDoulogos on Thu Jul 06, 2017 5:16 am, edited 1 time in total.