Most insurance companies (at least the big ones) calculate an estimated minimum cost to REBUILD. What goes into the REBUILD estimate? Cost of demolition and debris removal. Cost of permits and planning and architecture, costs for materials and labor to make your house what it was before. All those line items will go up in price each year, with no correlation to what the local housing market is doing. The insurance companies have no reason to care about the market value or tax value of your home. All they are trying to do is pay what the contract states, which is usually to make the home "whole" again.
The conversation becomes: is it worth it to me to insure this $80k house when it would take $120k to rebuild? That's a personal question. Without insurance, I'd say you'd be opening yourself up to big liability risks, like someone mentioned before, even if you could stomach a total loss.
Your best bet may be to find a broker, ask for minimal coverage for the home with a high deductible, but with solid liability coverage. The big companies probably won't be able to offer a good product that meets that need, but perhaps a local niche insurer will be able to offer that.