What happens to time when heat is added

Heat is energy causing particles to jiggle very fast. Assuming that rule stays constant no matter the amount of heat under planck temperature, does that mean that particles will eventually jiggle near the speed of light?

Also, because cold is the absence of heat, does that imply that cold things move faster through time than hot things?

In a sense, yes. The proper time "experienced" (not really experienced, of course, since atoms don't experience anything) by each atom should be a bit less when they're moving around more. At least in a classical picture where the atoms follow well-defined trajectories, which is true enough in an ideal gas. The object as a whole, though, is a different story--its center of mass stays the same regardless of all that atomic jiggling. So you have to ask yourself which reference frame you're talking about. Your question is perhaps more complex than you first thought!

Now if you really want some mind-blowing connections between temperature and relativity, look up the Unruh effect. It turns out that empty space has a temperature if you're accelerating.

No, really. Not kidding.

time is an independent variable and is not affected by heat.
and cold is an undefinable term and so is hot.
And now for something completely different.