The required temperature depends on the mass of the particles you're considering. You could say photons are always relativistic, so even the photon gas that is the cosmic microwave background is relativistic at 2.7 K. But you're presumably more interested in massive particles.

If you apply the kinetic theory of gases to hydrogen, you'll find that the average kinetic energy will reach relativistic levels (taken to be when it becomes comparable to the rest mass energy) around 10^12^ K. For the free electrons (since we'll be dealing with plasmas at any sort of relativistic temperatures), this temperature is around 10^9^ K due to the smaller mass of the electron. These temperatures are reached at the cores of newly-formed neutron stars (~10^12^ K) [1] and the accretion disks of stellar-mass black holes (~10^9^ K) [2], but not at the cores of typical stars. Regarding time dilation, an individual particle's clock would tick slower from the perspective of an observer in the center-of-mass frame of the relativistic gas, but I don't think this would have any noticeable effect on any of the bulk properties of the gas (except for the decay of any unstable particles). Length contraction would probably affect collision cross-sections, though I haven't done any calculations for this to say anything specific. One important effect would be the fact that the distribution of speeds would follow a Maxwell–Jüttner distribution instead of a Maxwell-Boltzmann distribution, and that collisions between particles could be energetic enough to create particle-antiparticle pairs. This would affect things like the number of particles in the gas, the relationship between temperature and pressure, the specific heat of the gas, etc.

You mention the early history of the Universe in your other comment. You can look through this table on Wikipedia to see the temperature range during each of the epochs of the early Universe, as well as a description of what happened. The temperatures become non-relativistic for electrons at some point during the photon epoch.

[1] https://doi.org/10.1063%2F1.4909560

[2] https://doi.org/10.1016%2Fj.isci.2021.103544

Kugelblitzes might (theoretically) be a thing...

To wit, a sufficiently dense concentration of heat, light, or radiation could produce an event horizon similar to that of a black hole, which definitely would count as a noticeable relativistic effect.

Based on one of your comments clarifying what you're wondering, I don't know that this helps you in what you're looking for, but the "OMG particle" came to my mind. It was traveling at such high energy when it hit our atmosphere that...

If the proton originated from a distance of 1.5 billion light years, it would take approximately 1.71 days in the reference frame of the proton to travel that distance.

...

The energy of the particle was some 40 million times that of the highest-energy protons that have been produced in any terrestrial particle accelerator.

...

In the center-of-mass frame of reference (which moved at almost the speed of light in our frame of reference), the products of the collision [with a particle in our atmosphere ] would therefore have had around 2900 TeV of energy, enough to transform the nucleus into many particles, moving apart at almost the speed of light even in this center-of-mass frame of reference. As with other cosmic rays, this generated a cascade of relativistic particles as the particles interacted with other nuclei.

I don't know if that cascade is the same as the Cherenkov radiation it produced, but that radiation is how they detected this particle, and it's interesting a.f.

[It is] emitted when a charged particle (such as an electron) passes through a dielectric medium (such as distilled water) at a speed greater than the phase velocity (speed of propagation of a wavefront in a medium) of light in that medium. ... Its cause is similar to the cause of a sonic boom....

~~I.e., (layman's understanding here) the particle, having a dual particle- and wave-like nature, is propagating through the vacuum of space "close" to the max speed of propagation of causality itself. As it encounters a medium, our atmosphere, it is going faster than causality itself can possibly propagate through that medium. But the energy is still there and isn't going to just vanish, so it has to split out into multiple particles that would, with their fraction of the original energy, then be able to propagate through the medium. Or something amazing like that?~~

Edit: My layman's understanding of Cherenkov radiation requires a bigger disclaimer, like a strike-through. :)

There are places in the universe that are so hot that weird things start to happen. Like the core of Jupiter could be a giant hard hydrogen crystal or in the center of suns where lighter chemicals fuse into heavier ones. Or my favorite, the temperature of the early universe which may have contributed to hyper inflation which would constitute what you refer to as "relativistic effects".

In terms of noticing them we have detected the cosmic microwave background radiation.

There's also a Planck temperature, which is the highest we can currently predict in the Standard Model - that's the temperature at which thermal radiation is at the highest possible energy

It's an interesting question, but a bit vague. Even at room temperature, relatively needs to be considered for the motion of electrons.

You're probably thinking about bigger stuff though. The short answer is that temperature is unbounded so yes, there is a temp at which it is significant for the motion of all particles. I think inside of stars this can happen, but my knowledge jn that area is pretty limited.

Veritassium has a recent video about some of this that you may find interesting if you haven't already seen it.