If there were lots of geothermal plants—even mobile ones—near a volcano, how much power could this provide? Could the sapping of some of the heat energy make the volcano less likely to erupt?
The question is a little off the mark, as volcanoes are just the surface expression of a much larger system. A more general idea is to target shallow magma chambers, and the hydrothermal fluids they act upon. This is being done all over the world, in tectonically active areas where there is a lot of molten rock at reasonably shallow depths.
The Iceland Deep Drilling Project have accidentally created the first magma-EGS system. Read more: pangea.stanford.edu/ERE/db/WGC/papers/WGC/2015/37001.pdf
Volcanoes erupt due to increase in pressure, within the magma chamber (often 3-10 km deep), due to exolusion of volatiles. The magma chamber is usually heated from below. Geothermal plants in volcanic regions exploit heat from hot water in the "shallow subsurface", few hundreds of meters deep (and rarely exceeding 1-2 km). So as far as your second last question is concerned the impact on the volcano will be little as deep hydrothermal circulation can only transport so much heat.
There are many such plants around the world, e.g., Iceland, Philippines, Indonesia, New Zealand etc. However drilling in hot rocks has its own set of problems. It is also not easy to build large plants near volcanic areas due to lack of accessibility, infrastructure, market etc.