Honestly, you don't care what's going on with your neighbor's house. You just want it not to look like an eyesore while you're trying to sell yours. Right?
As for what's going on, your neighbors have abandoned it. It's either been foreclosed on or shortly will be. There's the answer to your wrong question.
The right question is: How do I eliminate the eyesore, and thus don't raise red flags with potential buyers?
That answer is simple: You get someone over there periodically to mow the grass, remove the weeds, etc. You pay for it. (It shouldn't cost that much to get someone out there to mow the grass once a week. Maybe even only once every 2 weeks.
Look: You can complain to your town or city and, if the problem's bad enough, the town or city will try to contact the owners (now the bank) and get them to do something. Nothing will happen. The town or city might actually come out, eventually, and mow the lawn and send the bill to the bank. That's a 1-time fix. It could take months to happen. (You feel like having that eyesore there for months?) You can contact whatever bank foreclosed and ask them to do it. They won't.
Bottom line is: To get rid of the eyesore in a timely fashion, you're going to have to do it yourself. Find a gardener or lawn mowing service. Get them out, and schedule something on a regular basis.
Will the "owners"--the bank--object? No. Will the town or city object? No.
You want to get rid of the eyesore and sell your home? Don't worry about what's going on over there. (I already told you what's going on.) Fix it.
Hope that helps.