I was thinking of replacing my broken power button in my CPU with a tactile switch connected to the power cord on the motherboard, but I was wondering if that would work without any harmful consequences for the computer or me?
Assuming that your PSU is an ATX one,
PS-ON is an active low signal that turns on all of the main power rails including 3.3V, 5V, -5V, 12V, and -12V power rails. When this signal is held high by the PC board or left open circuited, outputs of the power rails should not deliver current and should be held at a zero potential with respect to ground. Power should be delivered to the rails only if the PS-ON signal is held at ground potential. This signal should be held at +5VDC by a pull-up resistor internal to the power supply.
Source: ATX Specification - version 2.01
So if your PSU is indeed an ATX supply, PS-ON seems to be a high impedance input. In that case, and if the power switch is connected directly to your PSU, then there shouldn't be a problem pulling it to GND using a tactile switch.
However, the power switch might be going through some logic or buffering on your motherboard, so you might want to look up the specifications for your motherboard if you want to make sure it's safe. I suspect it will be safe, but without knowing your motherboard there's no sure way to tell.