When you heat something up, especially food which has already been cooked and allowed to go cold, you make it hot. Freda heated up a pie for me. = warm up
When something heats up, it gradually becomes hotter. In the summer her mobile home heats up like an oven. cool down
When a situation heats up, things start to happen much more quickly and with increased interest and excitement among the people involved. Then in the last couple of years, the movement for democracy began to heat up. = hot up cool off