The primary issue with over voltage is heat. Most power supplies have some form of regulation built in to ensure the output voltage is constant. Any extra voltage is usually dissipated as heat, the more to regulate, the more heat is generated. This is especially true of older linear regulated PSUs.
Switch Mode Power Supplies (SMPSs) are less prone to this by the mere nature of their design. However, they are not totally immune to over-voltage. Apart from components being stressed beyond capability and being destroyed (covered in the section "component failure"), they are also prone to malfunctioning when operated outside the switching oscillator parameters.
Over voltage forces the oscillator to 'squedge'. This is usually through the regulator circuitry having to over compensate against an oscillator that can no longer be regulated at the level required, so the oscillator simply turns off. The output voltage falls, the regulator releases any control, and the oscillator starts again - and so does the cycle.
The oscillator circuit does not necessarily have to squedge but may reach a point where it can no longer regulate the output (either permanently or as the ripple voltage goes through peaks) which could manifest itself from mere disturbances to any circuitry it powers though to destruction of this circuitry.
Another effect of higher voltage is the current into the PSU drops and this could fall below the "maintenance current" of any NTCR (inrush protection) fitted. The effect noticed on one computer monitor was the screen became unfocused, then back into focus, the cycle lasting a few seconds. This was the NTCR becoming a high resistance creating a high voltage drop forcing the PSU to draw more current which warmed the NTCR thus allowing more voltage into the PSU which now had a lower current draw......