A point of correction – me being pedantic.
Mine shafts on earth are limited in how deep they can go by two
things, internal heat and rock pressure.
This is not correct. Remove the word shafts and yes, you are correct, the depth to which mines on Earth can go is limited by geothermal heat and rock pressure/stresses.
When it come to the limits of mine shafts, the length of an individual shaft is limited by the material properties of equipment used in the shaft, particularly if it is a hoisting shaft. The critical components of a shaft hoisting system are the ropes that move the skips and any counter weights and the drum of the winder on to which the rope is wound.
Due to self weight of the ropes and the internal flaws they posses, due to manufacturing processes, the maximum length of a hoisting shaft is about 2000 m. If a mine needs to go deeper, transfer tunnels need to be developed at the bottom of the first hoisting shaft and then another shaft is sunk from the transfer tunnels, to go deeper. This means, that in such a mine everything that enters and leaves the mine must be double handled: down one shaft, along the transfer tunnels and then down the other shaft and the reverse when going up.
If we accept the depth, due to heat and stresses, in one of the other answers, 24 km. A mine that deep on the Moon would have a series of twelve cascading hoisting shafts, each with their own system of winder and ropes. That appears overly complex to me and I doubt it would be practical.
An alternative to a hoisting shaft is a descending spiral tunnel, called a decline. A straight decline can be used, but a spiral contains the tunnel to a limited lateral region. Currently, on Earth, declines are being established with maximum gradients of 1 in 6. To get to a depth of 24 km, a decline would need to be 144 km long, horizontally (146 km slope length). That’s a long way to haul dirt out of a mine! Again, this appears impractical.
The added complication to this is, what form of tires would any trucks on the Moon have? NASA reinvented the tire for the Apollo lunar rovers. Can a similar type of mesh tire be made for heavy duty trucks hauling rock out of a mine on the Moon? If not, what type of tires or wheels would trucks on the Moon have? Would an inclined railway be required, or a conveyor belt? If a 146 km long conveyor belt was used, from what material would the belt be made of? How would spillage from the conveyor belt be cleaned up? Railroads using unassisted locomotives need shallower gradients which would mean their tunnels would be even longer. Assisted locomotives, such as rack railways use a third rail which have a toothed profile and the locomotives have a cogwheel that are used on steep sections railroad.
There’s more to a practical mining depth than just geothermal heat and ground stresses.