Why are the bees dying after eating aphids from trees?

Firstly, bees don't eat aphids: they are strictly vegetarian and eat nectar, pollen and honeydew, and it is the honeydew that is the problem here. Aphids feed on plant sap by puncturing the leaves with their mouth parts. Sap is mainly water so they need a lot of sap to get all the nutrients they require, and they excrete the surplus as honeydew. This is what the bees feed on.

Unfortunately certain trees, such as lime, have narcotic or toxic chemicals in their sap which don't affect the aphids but when concentrated in the honeydew will stupefy or kill the bees.