Hello. My question is about semen production in men. How long does it
take for the male body to replenish enough semen for ejaculation? Will
it run out at some point? If a male increases the frequency with with
he ejaculates - will the body compensate by producing more - thus
increasing volume even when it's done more frequently? Is there
anything unhealthy about doing so? Thank you.

The "recharge" period varies between people and with age, but a few
hours to a day is probably about typical (that'd be for a decent
ejaculation, not just a dry orgasm). The reserve of semen may run dry,
but it's replenished fast; males never permanently run out of semen.

You should change "never" to rarely. Some males save some for later,
but 4 times in the same hour likely means little or none the 4th
ejaculation.