Yes, the donation of an organ from a person with a food or medical allergy can cause allergies. Several studies have noted a correlation between allergies in donors, and recipients after receiving the organ. This correlation now has a name, "Transplant-Acquired Allergy", meaning that it must be somewhat common. Scientists have been working on medicines and techniques to eliminate Transplant-Acquired Allergies, though.
Organ donors do not pass on allergies. The cause of allergies today lies in the gut of said allergic person, and the diet with which they poison said gut. Allergies are a symptom of the fast food industry, the sugar industry, and the all-you-can-eat epidemic sweeping the nation. Organs don't do that.
I have not heard of anything like that happening in organ donor patients. In order to pass on an allergy the donor would have to pass on its own imune system. Allergies are the body's immune system overreacting to small pathogens and foreign bodies. This kind of thing would not carry over to the recipient of a donated organ.
The biological basis for allergy is immune cells or antibodies reacting to a certain allergen within the body. These constituents are produced by areas such as the bone marrow. If a singular organ, such as a heart, is being transferred from one to another, the allergen-sensitive immune cells won't be made by it, and thus the recipient will not have acquired an allergy.