Options:

• Your dish is not actually as cold as you think it is: verify with a thermometer that the dish is actually colder than ambient temperature. Your hand is a terrible thermometer because it does not reach thermal equilibrium with the object you are trying to measure, and so it is sensitive to the thermal conductivity of the substance you are touching.
• Your dish is resting on a surface that is colder than ambient temperature. I'm not sure what your dish is resting on. If you confirm that your dish is in fact colder than ambient temperature, you should check the surface it is on (again, with a thermometer, not your hand). Also check the temperature of the air.
• Your dish has a refrigerator hidden inside it. I say this partly as a joke, but it is plausible that there could be some endothermic chemical reaction going on. Cooling (below the surrounding temperature) requires work, which means the energy has to come from somewhere. This is why you have to plug in your fridge. For your plate, the only plausible energy source I can think of is chemical, though I've never heard of anything like this for a plastic dish before.
• Your dish is violating the second law of thermodynamics. This is improbable...

Options:

• Your dish is not actually as cold as you think it is: verify with a thermometer that the dish is actually colder than ambient temperature. Your hand is a terrible thermometer because it does not reach thermal equilibrium with the object you are trying to measure, and so it is sensitive to the thermal conductivity of the substance you are touching.
• Your dish is resting on a surface that is colder than ambient temperature. I'm not sure what your dish is resting on. If you confirm that your dish is in fact colder than ambient temperature, you should check the surface it is on (again, with a thermometer, not your hand). Also check the temperature of the air.
• Your dish has a refrigerator hidden inside it. I say this partly as a joke, but it is plausible that there could be some endothermic chemical reaction going on. Cooling (below the surrounding temperature) requires work, which means the energy has to come from somewhere. This is why you have to plug in your fridge. For your plate, the only plausible energy source I can think of is chemical, though I've never heard of anything like this for a plastic dish before.
• Your dish is violating the second law of thermodynamics. This is improbable...