Short answer: storing food in an open can is normally safe for a short period, but inadvisable.
There are three main issues with storing foods in an open can. They are metal oxidation, contamination, and funk.
(1) Oxidation. As commenters have observed, cans used to be made of tin, which is toxic. Modern cans are made from either steel or aluminum. If the contents of the can are acidic, cans are lined with a polymer (plastic) inner layer. If that layer is broken -- say, by a major dent in the can or by using a knife to scrape out contents -- then the acid can get to the metal and, over time, corrode it, and some of the now-oxidized metal will dissolve into the food. As far as we know, the oxidized metal is not toxic, but it tastes horrible.
(2) Contamination. Any open container of food can become contaminated when an airborne water droplet (tiny -- microns wide) containing a nasty microbe drips or settles into it. The major biological culprit here is Listeria, which can grow in any moist environment, including spaces refrigerated below 40F (5C). Also, if juices from your raw meat drips onto an upper shelf, which later gets wiped inadvertently into your container of ready-to-eat food, you are likely to get Salmonella or E coli. This risk can be mitigated if you keep your raw meats on the lowest shelf, but not eliminated. Also, it is worth noting that contamination through this pathway is a relatively rare event -- but even so, it is not worth the risk.
(3) Funk. Have you ever put an ice cube in your beverage and notice your beverage tastes strange? The chemical compounds that give food their flavor tend to be volatile (meaning they will readily leave the food into the surrounding air). Also, the mustiness of a refrigerator/freezer is due to volatile compounds produced by mold and mildew. Foods that are wet and/or fatty can be molecular velcro to these compounds.