I've rescued computers and CRT TVs/monitors that have been left out in the rain, I'd say that rust is the main enemy. Edge connectors and component legs will rust if not dried soon enough, causing intermittent connections, short circuits or even breakage (when component legs are so brittle that they break off just by touching them I consider it not worth repairing), but most things survive just fine if left to dry as soon as possible.
Where I live the rain is dirty because of the high air pollution and thus leaves a very recognizable layer/pattern of filth on cases and circuit boards, making it evident to the trained eye when something has been rained on. Anything you can do to speed up the drying process (disassembly, heat, air circulation, etc) will be beneficial. Non-sealed mechanical components will need special attention, and so do high voltage electronics like CRTs - if water gets into the tube socket it can cause bad focus (usually accompanied by the sound of frying bacon) and internal arcing leading to flashes, shutdowns or blown components. The tube socket is difficult to dry naturally, I usually spray a bit of contact cleaner on the pin holes to displace any trapped water followed by application of a hair dryer. If arcing continues, the socket could be internally rusted and might need to be replaced.