When you say you are ’turning up the heat’ it means you are putting more pressure on someone to do something or beginning to do something with more intensity. When you ’turn up the heat’ you make a situation more serious or difficult. This is an idiomatic expression used in both social and professional contexts.

1. Alex had owed me money for months. I only managed to get it back by turning up the heat on him.
2. We have been training really hard this year. We want to turn up the heat on our rivals in the championship.

1. The project team really turned up the heat as the deadline approached. It was a terrific team effort.
2. We're going to have to turn up the heat on our suppliers. The number of late shipments is unacceptable.