What type of circuit is responsible for convolution in the classic analog telephone?
Convolution is a mathematical abstraction describing how a linear, time-invariant system affects a signal going through it.

Sometimes one explicitly designs a system to convolve a signal by a predefined impulse response (for example when building a digital filter); but more often than not, convolution is used to model various physical processes involved in a system. These physical processes can be transmission delays or dispersions, the limited bandwidth of an amplifier, transmission medium or electro-mechanical system, the explicit use of passive R, L, C network to achieve some filtering, etc... Thus, it might not be out of place to find convolution used in the description of some elements of a telephone system, but it's a modeling tool, not an actual process.

Here is an analogy: the trajectory of a cannonball is a parabola, but it makes little sense to ask which mechanical device in a cannon is responsible for computing $y = ax^2 + b$.