Let us start from definitions, as they are given in many sources.
A quasistatic process is an idealized processes that happen out so slowly that a system go through a sequence of states arbitrarily close to equilibrium.
An isobaric process is a process, in which the pressure remains constant.
An isochoric process is a process, in which the volume remains constant.
An isothermal process is a process, in which the temperature remains constant.
Now, I could imagine how an isochoric process could not be quasistatic, but what about isobaric and isothermal processes?
For instance, for the following specific example (from Count Iblis answer):
Suppose we put a mixture of hydrogen gas and oxygen gas in a
conducting cylinder that is kept at constant volume and in thermal
contact with a heat bath. There is also a device in the cylinder that
will produce a spark, igniting the gas.
I indeed object (as Count Iblis expected) that the process of ignition is isothermal: when we ignite the mixture chemical reaction takes place, the huge amount of heat is liberated, and besides, a complicated movement of the gas mixture occurs. Thus, it does not make a lot of sense to speak about keeping the temperature of the system constant, since the temperature is different even in different regions of the system. This should not surprise us, since right after the ignition and for some time then the system is clearly not in equilibrium, and thus, the notion of the temperature of the system is not even defined.
So, if we adopt this definition then, an isothermal process imply a quasistatic one, since it does not make sense to talk about constant temperature in a process that is not quasistatic, since temperature is not even well-defined in intermediate non-equilibrium states of such process.
In other words, the presence of the heat bath is not enough for the process described above to be isothermal: the process should also be performed slowly enough so that all heat liberated can be almost instantaneously transferred to the surroundings. Only in such a case the temperature of the system is indeed constant during a process. Clearly, the system should be in thermal equilibrium with its surrounding all the way, and thus, I think that for the process to be isothermal, it should be quasistatic.
Am I right?
P.S. Oh, how I feel like I see where I went wrong. An isothermal process indeed must happen at such a slow rate that the thermal equilibrium is maintained, however, thermal equilibrium is not the whole story. As we know systems in thermodynamic equilibrium are always in thermal equilibrium, but the converse is not always true. So, since only thermal equilibrium is required for an isothermal process, it is not necessarily quasistatic. A quasistatic process is always arbitrarily close to thermodynamic equilibrium, while isothermal is only in thermal equilibrium.
Best Answer
No, not necessarily. The definition you gave for a quasi-static process is correct, and it clearly states "close to equilibrium", meaning that all three $P, V, T$ parameters should undergo infinitesimal changes, for the process to be considered quasi-static.
Whereas in either isobaric, isochoric and isothermal processes, each time only one of the 3 quantities are held constant, and neither of them will dictate whether the remaining quantities should evolve infinitesimally.
Of course it becomes very simple to understand when you just consider a set of examples where iso-processes occur far from equilibrium:
In Otto cycle: for example after the adiabatic compression, the follow-up combustion is considered isochoric wherein the pressure P shoots up, cannot be quasi-static.
Stirling engine: cycle: an isothermal expansion followed by an isochoric heat-removal, then an isothermal compression followed by an isochoric heat-addition, and all these transformations that the gas undergoes are non-quasistatic, meaning the energy changes of the system either due to heat exchange or work, do not take place in infinitesimal amounts.
EDIT: Further explanations:
Quasi-static processes: