Temperature itself is, seen from a fundamental point of view, a statistical quantity. It describes the part of the mean energy of a particle within a many-body system (fluid or solid) which is due to it's irregular motion in the local rest frame of the bulk medium.
The heat equation is, as any physical equation that describes the time evolution of systems, causal, i.e., it describes the evolution of a quantity (or several quantities if you have a coupled set of equations like Maxwell's Equations for electromagnetism), given the history of the quantity (or quantities) at times prior to the time in question. BTW: It is important to note that by definition, time is a directed quantity, i.e., the causality principle defines a "fundamental arrow of time", from which also other "arrows of time" may follow, e.g., the "thermodynamical arrow of time" according to Boltzmann's H theorem.
Very often, and for sure on the fundamental level, the equations are not only causal but even local in time, i.e., one doesn't have to know the whole history of the system prior to the time under consideration, but it is sufficient to know the state of the system at one point in time in the past. That's also true for the heat equation: If you know the temperature field at one point in time and appropriate boundary conditions, it provides a unique solution for the temperature field within the body at any later time.