roots go to moisture. they will go beyond the drip line in search of water. your irrigation system is inviting the roots over for a drink. i suggest you do the same thing with your nieghbor and see if he is willing to help you control the roots. you will need to dig a trench along the property line between your system and the tree. the trench should be about 14-16 inches deep severing the encroaching feeder roots. then use a barrier material like plastic too keep the roots out of your yard. the roots will turn at the plastic and search for moisture elsewhere.
i suggest renting a trencher to do the work. trying to do it by hand would be very difficult. be sure to locate your irrigation lines and any other utilities, or the trencher will.