Perhaps you're referring to the CPU fan wires that go from the heatsink fan to the motherboard - if so, an easy trick for that is to tie loose half hitches in the wire (two or three) to shorten it up and/or pulled taught (not tight!) and the excess tucked behind the top of the mobo (usually 20-22ga wire, should tuck in easily enough).
As far as wires touching the mobo consider that there are a few hot areas on the motherboard (CPU being the hottest), wires touching on those surfaces can prevent heat from dissipating (depending on size and surface area), also, there are minute vibrations going on in your PC, eventually the insulation could get worn through (notably if touching anything sharp like a heatsink fin) - that would be bad
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If you have a super cheap power supply (400W, came with $30 case), you might want to turn it off when not in use - or make sure your fire alarm batteries are fresh - seriously, no real reason to turn off the PSU itself afer the computer shuts down.