Road associations on abandoned roads are problematic for a number of reasons, liability being one of them. For readers not familiar with the situation, roads abandoned under Maine statute remain "public easements" - roads over which the public still has a right of travel although the public has no further maintenance obligation and no liability. That's one of my biggest disputes regarding abandoned roads.
How is it constitutionally sound to compel private individuals to maintain a road for public use at private expense? Yet if you do not maintain it, you eventually lose access due to use by abutters and the public, and you are prohibited from restricting public use of the road.
I am not as familiar with non-profit associations as I am with statutory associations. But I don't think there is anything that would prohibit you from posting such a sign, and it might reduce your liability somewhat. If nothing else, it might make people think twice before entering the road.
I have debated the possibility of various signs over the years. One I've thought of is, "Road Not Maintained at Public Expense." I don't think that could be disputed, as it is true. Another is, "Road Not Maintained for Public Use." That one might be disputed, even though it is true. The road is maintained for the abutters by the abutters, but public use cannot be prohibited due to the public easement that remains on an abandoned road.
A longer sign I have recommended is, "Warning: Road Not Maintained at Public Expense. Damaging a public easement with a motor vehicle or ATV is a Class E Crime." While that doesn't directly address liability, it might make people think twice before using the road, especially in mud season. You could also cite the relevant statute in fine print at the bottom, in case anyone questions it. (17 MRS 3853-D)
One other note about road signs: Under E-911 guidelines, road names on town ways are supposed to have white lettering on a green background. Road names on private roads are supposed to be white lettering on a blue background. Some towns get it right, but many do not. Even if they did, most people would not realize that a blue sign indicates a private road. And E-911 guidelines do not have any special sign color for abandoned roads that remain public easements.
You could ask the town if they have any objections to your posting your sign - point out that it could help protect the town as well. They may say that they already have no liability for the road, but you could counter that not everyone knows that, and they could still have to defend against a lawsuit if someone got injured. A warning sign could help ward off problems before they arise.