If someone left fully cooked food in an oven before Shabbos and left the oven on (a normal setting, not a Shabbos switch). What would be the problems if any of opening the oven to take out the food on Shabbos?
Opening the oven will involve a problem of cooling down the oven, thereby activating the thermostat, which will cause the oven to go on.
Although poskim are more lenient concerning the opening of a fridge (see Minchas Shlomo 1:10), the Shemiras Shabbos (1:35) is stringent concerning an oven, which is possibly more sensitive to opening the door, and whose activation involves a Torah prohibition of mav’ir.
However, Iggros Moshe (Orach Chaim 4, no. 74, Bishul 28) is lenient even concerning an oven, because it is only a safek pesik reisah (opening the door might be a pesik reisha of activating the oven, but it might not, in cases where the oven would anyway have been activated), and it is only a rabbinic prohibition (because it is considered a gerama).
It is right to act stringently, and there if an oven is left on, one should be careful to open the oven when the oven is already activated, so that opening the door will not have any effect. Under extenuating circumstances, one can rely on the opinion of Rav Moshe.
Note that the above applie specifically to regular ovens that do not have extra features. Some ovens have special features, such as extra light being turned on upon opening the door, or a turbo fan being turned on, and so on. For these ovens, you can contact the Institute for Technology and Halachah (at 02-6424880 (or fax: 02-6420949), 1 Hapisga St., Jerusalem), who will be able to advise further.