The statement "heat rises" is a very true statement, but when we say this we should think about ."which direction is "up"? Do we mean up away from the ground, or up north? It turns out that heat rises in both of these directions!

Heat rises due to density. Warm air is less dense (less heavy) than cold air. The cold air sinks and the 'lighter' warm air rises.

Let's look at how heat rises north.
The equator gets more sunshine and warmth from the sun. As you move away from the equator the sun rays aren't as strong, so as you move from the equator to the poles the temperatures get colder. So we can say that the warm air at the equator is lighter than the cold air at the poles. And in fact, air warm masses from the equator do move to the poles to mix with the cold air there.

"Up" means two different things in your question. When we talk about north and south, "up" means "towards the north pole." But when we say that heat rises, "up" here means "away from the ground."

Heat rises away from the ground anywhere on the planet: heat still rises away from the ground on the North Pole.

North is not up and south is not down. We draw maps that way, but only because it makes maps easier to read. In the past we drew maps so that east was up and west was down.

It's colder in the north because the angle formed between the ground and the sun's rays is smaller in the north than in the south. You can test this by holding a sheet of paper out in the day and seeing how the paper reflects less light as you tilt its edge closer and closer to that of the sun. South of the equator, it gets colder as you go further south, for the same reason. This is because the Earth is a ball, and not a sheet of paper, and the equator is where the Earth's surface is perpendicular to that of the sun's light.