Parkinson's disease is caused by the loss of dopamine production. Are there other diseases that affect different neurotransmitters similarly?

I would be cautious with saying that Parkinson’s disease (PD) is caused by the loss of dopamine production. The pathogenesis of PD is rather complex and not yet quite understood. But let’s say the majority of the symptoms in PD can be attributed to the degeneration of dopaminergic neurons in substantia nigra (SNpc).

There are quite a few diseases with disturbances in neurotransmitter production. Here are some examples:

Huntington’s disease is characterized by the degeneration of neurons predominantly in another part of the basal ganglia - the striatum, where neurons producing gamma-Aminobutyric acid (GABA) are preferentially affected. GABA is an inhibitory neurotransmitter, and reduction in its content results in uncontrolled movement, known as chorea.

Reduction in GABA-mediated inhibition also plays a role in epilepsy.

Acetylcholine (ACh) is a neurotransmitter implicated in learning and memory processing, and cholinergic neurons (among others) are severely affected by Alzheimer’s disease. 

 Perhaps most famously, reduction in serotonin signalling is a factor in major depression, evidenced best by the effectiveness of serotonin reuptake inhibitors in treatment of depression. 

Reference: Minja Belić