A dome is:

1. the best shape for taking stress from very heavy trucks putting all their weight on them without the manhole itself gradually bowing, and

2. is best at transferring that stress equally into the manhole wall (cast concrete cylinder) itself. (A square manhole + manhole cover would disperse force unfairly, potentially gradually cracking the manhole walls / requiring stronger walls. A flat circular manhole would disperse force upon the center of the manhole equally onto the manhole walls, allowing for lower-material-cost manhole walls. A domed manhole cover additionally disperses force from most points on the dome equally into the manhole walls — important, as vehicles won't necessarily be driving over the exact center of the manhole!)

...but really, this is the wrong direction to work in. The original reason manhole covers are round, is simply that the walls of a manhole are best made round, for the same reason drink cans and barrels are best made round: a closed cylinder is great at taking compressive force from a lid above; passing it through as soft, equal tensile force through its walls without buckling strain; and then turning that force back into an equal compressive force on the floor / subsurface.

Most manholes are generally small closed cylinders acting as maintenance areas for nearby pipes, with the pipes coming in through the sides of the manhole walls, and the concrete bottom floor of the manhole resting upon compacted earth.

In this situation, any shape for the manhole other than a cylinder — if driven over for years/decades by cars — would gradually pound the uneven force acting upon the manhole's floor into the earth below, unevenly accelerating soil subsidence. Eventually, you've created a sinkhole below the road, right outside the manhole wall on one side.