Does the fact that energy is not conserved in cosmology open the possibility of new matter/atoms being created in the universe? Does it open the possibility of new baryonic matter/atoms being created in the universe and avoiding the Heat Death of the universe?
 A: Almost certainly not.
At the moment the non-conservation of energy is mainly the production of dark energy. That is, since the dark energy density is constant (at least this appears to be the case) the expansion of the universe creates new dark energy from nowhere. If dark energy could convert to matter then this could create new matter. However there is no evidence, either experimental or theoretical, that this can happen.
Having said this, if any of the various theories of inflation turn out to be correct then a process very like this is responsible for all the matter in the universe. When the inflaton field decayed at the end of inflation the eventual end products of that decay were the particles we see around us today. However note that (a) the energy density of the inflaton field was vastly greater than the density of dark energy and (b) this is all highly speculative since we have no idea what the inflaton field was if it existed at all.
A: The answer is yes, even if we assume that there are no creation of matter directly from the dark energy. Although at the present cosmological epoch the amount of the matter that could be thus created is many orders below the threshold of detection. However the associated processes may become relevant in the very very very distant future of the universe.
The pathway is simple: cosmological horizon + quantum mechanics = matter creation. It is the same principle that is behind the Hawking radiation of black holes only on cosmological scales. And unlike Hawking radiation, which decreases the mass of black hole eventually leading to its evaporation, this particle creation is a consequence of unlimited accelerated cosmological expansion and would continue eternally.
The main driving force behind the accelerated cosmological expansion is the dark energy. If we assume that it is truly stable, that it corresponds to nonzero cosmological constant, then eventually the universe would enter the de Sitter phase where the causal patch of the universe would have a stable event horizon with a fixed temperature. If most of other matter inside this patch of the universe decays, most of whatever remained (such as electrons and positrons) are carried away outside  cosmological horizon by exponentially expanding universe, black holes evaporate, then at such late times, most of the content of any causally connected patch of the universe would be Gibbons-Hawking radiation at a fixed temperature $T_{\rm dS}= H_{*}/2\pi$, where $H_* = \sqrt{\Lambda/3}$ is the constant Hubble parameter and $\Lambda$ is cosmological constant. This radiation filling the universe is precisely the new created matter and it can potentially contain baryon matter including quite complex structures.
Of course for the majority of causal patches its contents would be rather dull: photons/gravitons of extremely large wavelengths and very very rarely occasionally some massive elementary particles. But, as the universe continues its exponential expansion number of such patches would continue to increase. And while the probability of any nontrivial contents in any given local patch remains very small, the overall number of 'tries' would continue to grow. So if we wait long enough, among the matter created in this universe could be sentient observers (so called Boltzmann brains). For example, here there is an estimate of a probability for the appearance of a Boltzmann brain as a result of fluctuation: $\exp(-10^{42})$, so the likely time for the first appearance of the Boltzmann brain would be $\exp(10^{42}) \,\text{Gyr}$.
And if our universe would exist for unlimited amount of time in the future, then most of sentient observers would be arising from such fluctuations. A lot of people seems to find this (potential) situation disturbing:


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*Page, D. N. (2008). Is our universe likely to decay within 20 billion years?. Physical Review D, 78(6), 063535, doi, arXiv.

*Bousso, R., & Freivogel, B. (2007). A paradox in the global description of the multiverse. Journal of High Energy Physics, 2007(06), 018, doi, arXiv.
Such a future state of the universe strictly speaking could not be called a 'heat death' since there is nonzero particle creation at constant positive temperature and since if we wait for long enough time we could observe arbitrarily large fluctuations, however for most of the time almost every causal patch of the universe would be almost empty (compared with the present day universe), so from the point of view of present day life, this state could be called a 'heat coma'.
Of course, at present, the temperature associated with cosmological horizon is many orders of magnitude smaller than the temperatures of supermassive black holes not to mention the temperatures of many others astrophysical subsystems, so any matter created by this mechanism would be drowned by the noise of many other processes happening now, so the times when these effects could become relevant are in a  very distant future.
A: In this answer, I'll discuss the global average state of the matter in the universe, as described by cosmological models such as lambda-CDM models, which assume homogeneity and isotropy. That means that I'm not discussing extreme fluctuations or things like Boltzmann brains, which AVS's answer talks about.

Does the fact that energy is not conserved in cosmology opens the possibility of new matter/atoms being created in the universe?

Creating new baryonic matter is something that was considered seriously in the context of the steady state model, which was advocated by Hoyle, Bondi, and Gold up until about the mid-1960's, when evidence for a hot big bang killed the steady state model off. One of the problems with steady state models was that they had to violate Lorentz invariance, since there had to be some preferred state of motion for the newly created hydrogen atoms. (They also violate the charge conjugation and time reversal symmetries.)
You're right that mass-energy is not globally conserved in general relativity, but it is exactly locally conserved in the sense that the divergence $\nabla_a T^a{}_b$ of the stress-energy tensor is zero.
Our universe is currently pretty well approximated by the de Sitter spacetime, and is expected to remain so forever, according to current theories. For simplicity, let's suppose for the moment that the geometry of the universe has to be exactly de Sitter. This was what Hoyle et al. wanted, because they wanted all eras of the universe to look the same, and the de Sitter universe is the only cosmological model that has this symmetry. This seems pretty similar to the physical motivation for your question, which was whether it would be possible for the universe to avoid a fate in which there was basically nothing there but dark energy.
For the de Sitter spacetime the divergence of the stress-energy tensor has a timelike component equal to
$$\frac{\dot{a}}{a}(\rho+P),$$
where $a$ is the scale factor describing cosmological expansion, $\rho$ is the mass-energy density, and $P$ is the pressure. (This is all in units where $c=1$.) For this reason, we must have
$$\rho+P=0$$
everywhere. Dark energy satisfies this condition, but baryonic matter doesn't. Therefore it is not possible for new baryonic matter to be created in cosmological expansion, in the de Sitter spacetime.
If you're a wily theoretician like Hoyle and you look for a way to escape this constraint, there is a way out, which is to posit the existence of a field with $\rho=0$ and $P<0$. Hoyle called this the C field. Then if you add the contributions to the stress-energy from the C field and baryonic matter, you can end up with $\rho+P=0$.
Although dark energy is currently the dominant form of mass-energy in the universe, realistic cosmological models do incorporate other matter fields, including baryonic matter. These models therefore do not have exactly the geometry of de Sitter space. That complicates things compared to the argument given above, but the conclusion is still the same. According to these models, you can't have the production of new baryonic matter without violating local conservation of mass-energy, which is baked in to the structure of general relativity.
If you try to construct a model that produces new baryonic matter without violating local conservation of mass-energy, then as far as I know you are uniquely led to something like Hoyle's "C-field" theory, and then you have all the problems of that theory, including violation of Lorentz invariance and incompatibility with observations such as the cosmic microwave background. For more information on (failed) attempts to reconcile such theories with modern knowledge, see this web page by Ned Wright. I also have a mathematical discussion of the steady-state model in section 8.4 of my own GR book.

What raised my question is the idea of Eternal Inflation. New matter is being created in the newly created bubble universes. Of course this is highly speculative but as a layman I wonder if it’s possible since energy is not conserved. 

Energy is locally conserved. It's just not globally conserved. John Rennie's answer explains why inflation isn't an exception to this.
A: Of course John Rennie's answer is right, I would just like to add a few things.
New matter was created during the pair-production in the early universe with another happening called baryon asymmetry.
Now in terms of creating matter, pair-production is happening continuously in the Universe, just think about a neutron, inside it the sea of quarks, where quark-antiquark pairs are being created, and annihilated.
It is baryon asymmetry that we do not observe anymore in the universe, and without it, there is no new matter (normal matter) being created as far as we know.
We do not know about dark matter if it is being created. It is dark energy that is being created in the expanding universe with constant density.
Basically, for new matter to be created, baryon asymmetry would need much much higher energy levels that we have right now.
Please look at this question and the answer:
Baryon asymmetry still going on

Baryon number violation, one of the requirements for baryon asymmetry, only occurs at any significant rate in the Standard Model at high temperatures, much higher than are known to exist in the universe.
We know experimentally that any other processes that violates baryon number must be quite rare and/or occur only in extreme conditions, since we have not yet observed any.
Theoretically, black holes violate baryon number conservation. Black holes don't have a baryon number, and so when a baryon falls into a black hole, its baryon number is lost. If you consider a neutron star collapsing into a black hole and what happens afterwards, its pretty easy to convince yourself that baryon number can't possibly be conserved.

