To my knowledge, the single problem that you might run in to is discharge of the CMOS battery. In modern systems, the CMOS is maintaned by the 5vsb from the power supply whenever the computer is plugged in, so the CMOS battery is only required when the machine is completely unplugged. And, of course, the only bad thing that happens when the CMOS battery dies is that you lose your BIOS settings, which are probably mostly all defaults anyway (you will need to set the clock again).
In older systems, the CMOS battery was used whenever the system was turned off (because of how AT power worked) and sometimes had a very short life, so the system needed to be turned on semi-frequently to keep the CMOS charged up (much like how a car needs to be started to charge the battery, although the idle drain on a CMOS battery will kill it faster than a car battery - so think a car battery when you never turn your dome lights off). But honestly I don't think this is a problem you're going to run in to in a consequential way on modern computers.
Perhaps mechanical parts could seize up as a result of non-use (like the hard disk rotor or heads), but I've never heard of this happening and I think those devices are carefully enough engineered that it won't.
So really, the only reason I can think of to start a computer periodically is to keep up on updates, just so that next time you want to use it there won't be 100+ updates to install.