A cabin crewmember here.
The answer is: No, it is NOT allowed!
For various reasons, including:
- Security: We do not know what are you heating, is it a heat-activated bomb? is it something that might make a lot of smoke? etc.
- Health: We cannot contaminate the ovens, we know the source of the food that is provided by the airlines, we do not know the source of your food. Same ovens are used to heat the pilots' meals as well, we can't afford to have one of them having food poisoning in the middle of the flight?
The same goes for the refrigerators also.
Cabin crew do help passengers when it comes to babies' milk and so by providing hot water to heat up the bottle or a bag full of ice to cool things like medicines, but we shall never use the aircraft equipment to heat/cool any other food that is not provided by the airline.
There's a policy in the airline I work for that prohibits even the crewmembers from using the ovens/refrigerators to heat/cool their own food that they bring from outside the aircraft.
Some might say that they had a different experience and the crew helped them, yes it could happen when a crewmember decides to violate the rules to help a passenger based on his/her judgment, but that's not something to expect on every flight.
Factoids: there are various heating devices onboard airplanes, depending on the model and the airline, including Conventional ovens, steam ovens, microwave ovens, bun warmers, water heaters, steam wands, coffee makers, espresso machines, etc. Airlines and manufacturers have really taken these flying machines to the next level.