The WKB approximation states that in one dimension, the tunneling probability $P$ can be approximated as
$\ln P=-\frac{2\sqrt{2m}}{\hbar}\int_a^b \sqrt{V-E} dx$ ,
where the limits of integration $a$ and $b$ are the classical turning points, $m$ is the reduced mass, the electrical potential $V$ is a function of $x$, and $E$ is the total energy. Setting $V=kq_1q_2/x$, we have for the integral
$I=\int_a^b \sqrt{V-E} dx$
$=\frac{kq_1k_2}{\sqrt{E}}\int_{A}^1\sqrt{u^{-1}-1} du$ ,
where $A=a/b$. The indefinite integral equals $-u\sqrt{u^{-1}-1}+\tan^{-1}\sqrt{u^{-1}-1}$, and for $A\ll 1$ the definite integral is then $\pi/2$. The result is
$\ln P=-\frac{\pi kq_1q_2}{\hbar}\sqrt{\frac{2m}{E}} $ .
This result was obtained in Gamow 1938, and $G=-(1/2)\ln P$ is referred to as the Gamow factor or Gamow-Sommerfeld factor.
The fact that the integral $\int_A^1\ldots$ can be approximated as $\int_0^1\ldots$ tells us that the right-hand tail of the barrier dominates, i.e., it is hard for the nuclei to travel through the very long stretch of $\sim 1$ nm over which the motion is only mildly classically forbidden, but if they can do that, it's relatively easy for them to penetrate the highly classically forbidden region at $x\sim1$ fm. Surprisingly, the result can be written in a form that depends only on $m$ and $E$, but not on $a$, i.e., we don't even have to know the range of the strong nuclear force in order to calculate the result.
The generic WKB expression depends on $E$ through an expression of the form $V-E$, which might have led us to believe that with a 1 MeV barrier, it would make little difference whether $E$ was 1 eV or 1 keV, and fusion would be just as likely in trees and houses as in the sun. But because the tunneling probability is dominated by the tail of the barrier, not its peak, the final result ends up depending on $1/\sqrt{E}$.
Because $P$ increases extremely rapidly as a function of $E$, fusion is dominated by nuclei whose energies lie in the tails of the Maxwellian distribution. There is a narrow range of energies, known as the Gamow window, in which the product of $P$ and the Maxwell distribution is large enough to contibute significantly to the rate of fusion.
Gamow and Teller, Phys. Rev. 53 (1938) 608
In astronomy parlance, the Sun has a "metal"$^{1}$ mass fraction of about $0.02$. A solar mass is $\sim2\times10^{30}\;\rm{kg}$, so the sun contains about $4\times10^{28}\;{\rm kg}$ of "metals". That's about $20$ times the mass of Jupiter. A lot of that metal mass will be ${\rm C}$ and ${\rm O}$ and other elements a chemist would call non-metals, but I think there should be enough ${\rm Fe}$, ${\rm Na}$, ${\rm Mg}$, etc. to make at least a small planet or large moon.
The elements you drop into the star would be roughly sorted into concentric spherical shells, ordered with the heaviest elements in the middle, given enough time. There is a serious risk, depending on the masses and elements involved, that whatever you drop in starts fusing and making different elements, or if the right thresholds are exceeded, that the entire thing explodes in a supernova.
The Sun is a fairly typical star, not especially massive or puny, metallicity not remarkably high or low. I already showed that the Sun has a fair bit of metal without anyone dropping in any extra, and there are massive metal rich stars out there that have more than a solar mass worth of metals inside occurring naturally. In fact, very nearly every atom in the Universe that is not hydrogen, helium or lithium was made inside a star (and anything heavier than iron was most likely made when a star exploded). Some metals get ejected in supernovae and in stellar winds, but a large fraction of the metal budget of the Universe is already locked up in stars.
It would be possible to detect the contents of a star, with a carefully measured spectrum of the atmosphere and sophisticated stellar modelling (the spectrum serves as a boundary condition for the model). It would be more difficult than what astronomers do today since part of what goes into the models is guided by how metals are transported naturally in the Universe; artificially moving stuff around throws a wrench in the gears, but it's plausible that a concerted effort by an intelligent civilisation could develop the necessary science.
Olin Lathrop and John Rennie have raised some concerns about retrieval. I agree that a wormhole is probably a bad idea. Perhaps your best bet for retrieval is to set some carefully calculated extra mass on a collision course, stand (way, way, way) back, let the star go BOOM, wait a thousand years or so for things to cool off, then harvest the metals out of the gas of the nebula. A $1000$ year old supernova remnant is still pretty harsh conditions; the Crab nebula (exploded in the $11^{\rm th}$ century) has a temperature of about $10,000\;{\rm K}$, but rather low density. I'd call it plausibly survivable by a suitably advanced spacecraft.
$^{1}$Astronomers call everything that is not ${\rm H}$ or ${\rm He}$ a "metal".
Best Answer
No, I don't believe there is. Or, describing the scope of my answer, there is no maximum "metallicity" (for any normal mixture of metals) that could prevent a collapsing protostar becoming hot enough in its core to initiate nuclear fusion. (If your question is about the Jeans mass and metallicity, then you could clarify).
What determines whether fusion will ever commence is whether the contraction of the protostar is halted by electron degeneracy pressure before reaching a temperature sufficient for nuclear ignition.
For a solar composition protostar, the critical mass is about $0.08M_{\odot}$. Below this, the core does not attain a temperature of $\sim 5\times 10^{6}$ K that are required for nuclear fusion.
The calculation of this minimum mass depends on $\mu_e$, the number of mass units per electron in the core (which governs electron degeneracy pressure), and on $\mu$, the number of mass units per particle in the core (which governs perfect gas pressure). However, these dependencies are not extreme. In the core of the protosun, $\mu_e \sim 1.2$ and $\mu \sim 0.6$. If we made a metal rich star that had very little hydrogen by number and the rest say oxygen (a.k.a. a star made of water), then $\mu_e \sim 1.8$ and $\mu \sim 1.6$. The minimum mass for hydrogen fusion is given approximately by $$ M_{\rm min} \simeq 0.08 \left( \frac{\mu}{0.5} \right)^{-3/2} \left(\frac{\mu_e}{1.2}\right)^{-1/2}$$ (e.g., see here).
These different parameters would be enough to change the minimum mass (downwards actually) for hydrogen fusion to around $0.012 M_{\odot}$.
We could of course hypothesise a star that was wholly made of metals. A convenient estimate of the minimum mass for carbon fusion is already supplied by stellar evolution models. A $>8M_{\odot}$ star with a carbon core will initiate carbon fusion before it becomes degenerate. The mass is much higher than for H fusion because of the increased coulomb barrier between carbon nuclei. Of course the star also has a hydrogen/helium envelope, but if you replaced this with carbon, then the result will be little changed. Thus you could have a population of lower mass objects that do not become stable "stars". Those with masses of $1.4 < M/M_{\odot} < 8$ would presumably end up detonating as some kind of type Ia supernovae, because they will achieve a density/temperature combination where C can fuse, but in highly degenerate conditions. Lower than that and it becomes a stable white dwarf.
Of course your metal rich "star" could just be a ball of iron, in which case nuclear fusion isn't going to happen and if it is more than $\sim 1.2M_{\odot}$ it will collapse directly to a neutron star or black hole, possibly via some sort of supernova. Lower than that and it becomes a stable iron white dwarf.