Click the Whiteboard link at your house.
It doesn't show the exact percentage, but says stuff like "It's very cold outside" or "It's getting warm."
There's also the walkie-talkie icon in the lower left corner of the screen. It changes color from blue to yellow to red depending on how much heat you've got.
I once sold about 40 drug shipments to the Distributor at once and got something like 300-400 heat from it, but there was no instant arrest or anything. The icon turned red and there were signs of increased surveillance all over the different districts, but I got home and went to bed and the heat dropped just fine. The 404 Bleach service helped there too.