I have a neighbor who bags their grass and just stacks it up the back. They do not compose it. What problems can this cause if any? The trimmings are about 3 or 4 feet high now. Is this a problem for mosquitoes or any type of animals? Can this cause any type of fire?
This practice would definitely kill the grass underneath the bags of grass clippings. Whether it could cause any other problems would depend on the climate conditions and the way the grass is stored. If there is any moisture around, the grass will likely begin composting by itself if it is left in place long enough. This shouldn't cause any more problems than a regular compost pile, except for possibly being unsightly.
If you are in a very dry climate and the grass is left to dry, I suppose there is a possibility it could catch fire. However this would usually be a risk in grass that is purposely dried for use as straw:
Any rain or moisture on the grass would prevent this.