The probable reason why poeple say ice cream heats up your body is when you compare the amount of heat that it takes away your body compared to the food caloric value of the ice-cream.
100 grams vanilla ice cream is about 207 food Calories ($207 kcal$) according to this site.
On the other hand, the amount of heat that the ice cream takes from the body is roughly the amount needed to raise the temperature of the ice cream by $37C^0$ (assuming the ice cream is initially $0 ^0C$):
$$Q = mL_f + mc\Delta T = 0.1kg(44.46 kcal/kg) + 0.1kg(0.74kcal/kgC^0)(37C^0) = 7.184 kcal$$
(specific heat from this site)(latent heat of fusion from this site)
But of course, not all caloric content is absorbed, and according to this site, "Pure carbohydrate would leave you with a net 90-95 calories", which can be applicable to ice cream since it is mostly sugar (carbohydrate). So we are left with $0.90 \times207kcal = 186.3kcal$.
Also according to this site, "70-80% of the total amount of calories you burn each day is basically from thermo-regulation". This will leave us with a net $0.7\times 186.3kcal = 130.41kcal$ used for thermoregulation(heating up the body)
Not to mention that not all of this energy will be directly burned but some will be stored as energy (in the form of fats, etc.) as Steeven pointed out (and possibly some energy is used in the storing process, but I cannot find a source to figure it out, but these stored energy will eventually be burned when the body needs to anyway).
So, despite an ice cream initially cools your body when you eat it, it actually gives you more than enough energy to restore the heat it has taken away from your body. But of course, it does not necessarily mean that you'll end up a lot hotter than before you eat it. After all, our body regulates the temperature so that it is more or less $37^0C$ and reserves the rest of the energy currently not needed.