What physical evidence is there that subatomic particles pop in and out of existence? What physical evidence shows that subatomic particles pop in and out of existence?
 A: My current understanding is that the physical reality of vacuum fluctuations, particle-antiparticle pairs being created and then annihilating, is disputed. The Casimir effect is often cited as physical evidence but there's a few authors which have come to dispute that the Casimir effect is convincing evidence for the reality of vacuum fluctuations, as they argue that the same results can be extracted from treating the effect as a result of retarded van der Waals forces, not vacuum fluctuations.
See this paper: http://arxiv.org/abs/hep-th/0503158
and
this summary of the situation http://orbi.ulg.ac.be/bitstream/2268/137507/1/238.pdf
Maybe read the summary first, it's easy and quick to read :) As far as I know aside from the Casimir effect we have no other evidence for the physical reality of vacuum fluctuations. If you want to delve deeper, a good start is the papers cited by the two above.
A: $$\sin(x) = x-\frac{x^3}{3!} + \text{trigonometric fluctuations}$$
Above you can see why I don't like  the language of "quantum fluctuations" -- what people mean by them is just "terms in perturbation series we can't make classical sense of".   Similarly the phrase

... particles pop in and out of existence...

is a yet another naive attempt of describing quantum effects in a classical language. And there is no classical analogy that would reflect the quantum description of the world in a total accuracy.
On the other hand I cannot say that this language is wrong -- it is formally correct. Problem is that it just puts the cart before the horse, making a lot of unnecessary confusion. 
To sum up my answer: your question is wrong, since you are asking for "evidence" for a popular naive description of quantum phenomena in a classical language. 
What you should actually ask about is the experimental evidence for quantum mechanics and quantum field theory. And the experimental evidence for quantum description of the world is made of plethora of famous, not-so-famous and not-at-all-famous experiments. There are even mundane devices that exploit intricacies of quantum mechanics for the benefit of human beings. (I'm pretty sure that you can find those without my help.)
A: This phenomenon is called Quantum Fluctuations or vacuum energy and it could be described theoretically by Heisenberg uncertainty relation with the energy term. 
One of the physical evidences of such phenomenon is ''Casimir effect'' . 
when two uncharged plates are put close to each other they exhibit a repulsive force, this force is explained by quantum fluctuations (subatomic particles popping in an out of existence). 
A: This answer is basically an argument about why you should treat the terms of a perturbation series as interesting objects under the right circumstances. It doesn't really change the fact that these are just mathematical terms, but it shows that they have explanatory value in addition to simply being part of the sum because each term can be the leading term in the sum of another physical process.

Meson Production
A significant contribution to forward, production of pions and other mesons is the knock-on of quark-pairs from the nucleon sea. Reactions like
$$ e^- + p \to e^- + \pi^+ + \text{undetected hadronic junk} \,.$$
For one of many more technical set of discussions, see the $f_\pi$ collaboration's papers:1


*

*http://inspirehep.net/record/535171

*http://inspirehep.net/record/1290558

*http://inspirehep.net/record/1334567
Drell-Yan
The "Drell-Yan" process is
$$ q + \bar{q} \to l + \bar{l} \,,$$
with $q$ representing a quark and $l$ representing a charged lepton (experimentally one is generally interested in muons because the signature is experimentally easy to find).
It is obtainable in collisions between two protons. Protons have a valence quark content of $uud$. So where does the anti-quark in the initial state come from? From the nucleon Sea. Experiments using this technique include NuSea and SeaQuest

1 Chosen because I know which ones they are on account of having been part of the collaboration way back at the dawn of my career.
A: The best experimental evidence comes from photon-photon collisions. These occur during collisions of virtual particles (e.g. quarks) surrounding bremsstrahlung photons in an $e^{+}e^{-}$ collider.
From the tone of your question, I get the feeling you're looking for a simple, common experience example. However, the phenomenon is a high-energy quantum effect that is not obvious at the energy scales we live our normal lives. I don't think you'd notice it in a candle flame, for example.
At the energy scales of the LEP collider (on which I worked on in the 1990s), the effect was very real and created frequent events in our detectors that had to be corrected for.
Just as an aside, I'd be interested to know what aspect of Biblical Inerrancy you fear this phenomenon might challenge...
