The tree was trimmed once about 5 years ago, but now the neighbor says
money is tight and he won't do it again, even though he stated he knows
it needs trimming.


You could offer to pay for the trimming. If you pay they would probably
follow your wishes. The raining sap is caused by insects. You could offer
to have the tree sprayed as well. Legally you have no ability to make him
do anything so reach for your wallet.