Well first, what is a dedicated microprocessor? My guess it is a device that processes information - sort of like a mini-computer. The mini comes from the fact it is called "micro" processor.
So a product that has a dedicated microprocessor contains this little computer within it, specifically dedicated to performing the functions of that device.
My first thought was the health industry when trying to find examples, and I think I'm right. Devices that help doctors do things like produce analysis on blood flow or heart rate can have dedicated microprocessors.
So too can devices such as the one that controls urinary functions in people that suffer impairments in this area.
But what about industry and engineering? This field too has applications for microprocessors dedicated to purpose-designed tasks. For example:
LDV (laser Doppler velocimeter)
Controlling turbogenerators and semi-conductors.