I would like to prune my monkey puzzle tree, the bottom branches make it difficult to mow the grass. Is there a good time of year to prune? Will pruning affect the shape and growth of the tree?
Now is as good a time as any. You'll get lots of resin bleeding from the cut branch bases, that's normal. Pruning will slightly reduce the growth rate, particularly the trunk's diameter growth - the longer you can wait before doing it, the better it'll be for the tree (in a few years, the lower branches will probably start dieing off naturally, if you can wait that long, then that's best of all for the tree's health).
I've pruned the lower branches off of my monkey puzzle trees and noticed no resin bleeding or reduced vigor. Sure makes it a lot less painful for mowing!
Like other conifers more impressive with low branching. In fact, with this species specifically loss of lower branches indicates tougher, less favorable situations. If you make a lawn-free, mulched area around it that will eliminate conflict with the sharp leaves without diminishing the tree.
Won't make a lot of difference if it is the lowest branches, but removing whole branches might look a bit better.