Does Heat Cause Time Dilation?

No, the hot object itself experiences no relativistic effects. The particles in the object however may well experience time dilation or length contraction. In fact in very hot gases and plasma, ideal gas laws may not always apply due to relativistic effects.

These effects are especially noticeable in gases once the kinetic energy or fermi energy (for fermionic gases) exceeds $mc^2$ by a significant amount.

Relativistic Gas Problems

A hot gas will exhibit relativistic effects for its particles. A particular particle will have a time dilation effect between collisions with other particles. If its energy is $E = K + mc^2$, $K$ = kinetic energy, and mass is $m$ the Lorentz gamma factor for that particle is then $\gamma = E/mc^2$. This happens to all of the particles for their energy $E$ determined by a Boltzmann distribution.

What about the whole gas? Since the macroscopic object is stationary with respect to your frame there is no relativistic effect for the whole thing. In special relativity the Lorentz gamma factor is between the frame of a moving body and the frame of the observer. All of those gas particles are on different frames. They can't be said to define a relativistic effect as a statistical average.