Does a fast process always have to be adiabatic?

If a process is rapid enough that there is little heat transfer between system and surroundings then treating it as adiabatic serves as a good first approximation (an adiabatic process strictly requires zero heat transfer). For example this approximation is employed in calculating sound speed through a medium, because contraction-expansion cycle of the medium due to passage of acoustic wave is considered rapid. Ultimately whether such an approximation is good enough is verified only by doing experiments.

A slow process on the other hand approximates a quasistatic process. A quasistatic process is one in which the system passes through a succession of equilibrium states while executing a process. A quasistatic process can be isothermal if the process involves maintaining constant temperature, but is not limited to it.