But if heat consists of the speed of the molecule (which is an if) then shouldn't there be an Absolute Infinity as well as an Absolute Zero?
This question's "if" is not correct. Temperature (not "heat", as we use this word in a specific technical way) consists of the energy, not the speed, of particles. While these two are obviously related, it's important to note that while speed has a limit in special relativity, energy is unbounded. Energy approaches infinity (that is, increases without bound) as speed approaches $c$, so as you heat up the gas it can gain limitless energy without running into any problems due to the speed limit.
And are there quantum repercussions if some plasma approaches the speed of light through the individual speeds of its molecules?
You've got some answers about the Planck scale, but I tend to be pretty skeptical of these numbers--Planck unit sometimes smell of numerology to me, and it should be noted that the "Planck mass" is macroscopic and quite ordinary, so not everything that comes from mashing constants together gives you some amazing new physics. However, there are repercussions from having a gas at such high energy, and that is that new high-energy physics may become important, and we don't necessarily know what this is yet. Whenever you read about the LHC "recreating the Big Bang" or similar, this is because, in the early universe, temperatures were very high. Thus, particles that are now exotic or rare could be created from vacuum, such as heavy quarks, etc. As the universe cooled, there wasn't enough "excess" energy flying around to create these things. But conceivably, a really hot gas would have (for instance) Higgs bosons flying around in it. Until we have a complete theory of all particles, then, it's hard to say what starts to show up in the gas at high temperatures--but this is essentially what particle accelerators are trying to do. You do not run into a limit due to relativity on the temperature, however.
Does this also mean that the process of heating objects up is not linear anymore (because as you approach c, it becomes harder to accelerate the molecules)?
It wasn't necessarily linear to start with. The specific heat of an ideal gas is constant, but this isn't necessarily true for other models. For instance, at low temperatures (below what we call the Debye temperature), solids have a $C_v \propto T^3$.
Addendum: you're interested in the behavior of a gas of particles at high speeds. But all this really means is that the gas is ultrarelativistic, that is, $E \approx pc$ instead of $E \approx \frac{p^2}{2 m}$. Particles like this exist--after all, we study (extensively) the photon gas, and the model actually describes all particles that go fast enough to have the $E = pc$ dispersion relation.
