Viruses are software programs, and they can do the same things as any other programs running on a computer. The actual effect of any particular virus depends on how it was programmed by the person who wrote the virus.
Some viruses are deliberately designed to damage files or otherwise interfere with your computer's operation, while others don't do anything but try to spread themselves around. But even the ones that just spread themselves are harmful, since they damage files and may cause other problems in the process of spreading.
Note that viruses can't do any damage to hardware: they won't melt down your CPU, burn out your hard drive, cause your monitor to explode, etc. Warnings about viruses that will physically destroy your computer are usually hoaxes, not legitimate virus warnings.
