OK, speaking as a former Village President in a "country village", first, contact the Town Clerk to clearly understand the ordinance that prohibits burning. You need to be sure that the activity is clearly prohibited by an ordinance, then you need to understand if it is a Town ordinance or a county ordinance, as this will determine who is responsible for enforcement. Also, find out if your town or county has a nuisance ordinance.
If it is a Town ordinance, the Town Chairman and the Town police or constable, if you have any, are responsible for enforcement. If it is a County ordinance then the Sheriff is responsible for enforcement. The sheriff cannot enforce a Town ordinance.
Town ordinance- Call your Town Chairman and explain the situation, cite the ordinances, and ask him how he would like to proceed. Be nice, don't complain, just explain the situation and ask him what the next steps are, and that you would like this resolved in 30 days. Follow up with a congenial e-mail summarizing your discussion and thanking him for his help. If there is no action after 30 days (1 Town meeting cycle) file a written complaint to Town Chairman and Town Clerk, after 60 days have the next letter written by a lawyer.
If it is a county ordinance, get a copy of it and go to the Sheriff's office in person. Talk to the deputy on duty at the desk, explain the ordinance, and ask him what next steps are. Be polite, get his contact info and get the Sheriff's business card. Follow up with a nice e-mail summarizing and thanking the deputy for his/her help, copy the Sheriff and your county Supervisor on the e-mail. If no activity after 30 days, send a letter to the Sheriff and your County Supervisor directly. If no resolution after 60 days have a lawyer write the letter.
This all assumes the activity is illegal, which is why the "move the burning barrel" solution will not work. If it is illegal then he must cease and desist the illegal activity, and if he does not then he must face the penalties associate with it.
If there is no specific ordinance prohibiting this activity then the nuisance ordinance can be used to require a different solution, which may include moving the barrel or ceasing the activity.
As to talking to the neighbor, I would look at it this way. If the activity is illegal then don't confront him. It is no different than any other illegal activity and the enforcement body should get it stopped. If it is a nuisance activity then you are asking for a compromise and that could be discussed with him in person.
Just my humble opinion, and no I am not a lawyer and all other disclaimers are inserted here.<font color="ff0000">•</font><font color="ff0000">•</font><font color="ff0000">•</font><font color="ff0000">•</font>