When I was still at school, I had occasion to tour the local sewage treatment facility as a field trip. They had equipment to power all the buildings on site with captured methane, but did not use it, because it was cheaper to just buy electricity from the grid supplier than it was to operate and maintain the generator with free methane fuel. (The methane was still burned.)
The awful thing is that the greenhouse gas emissions involved in manufacturing the generator and maintaining it might have been significantly greater than the emissions due to the grid electricity. Not all these decisions are simple.
Better might be to capture the methane and use it to supplement residential natural gas. Here in BC you have the option to convert any percentage of your residential NG usage to 'biogas' AKA methane. (You just need to pay the higher cost for the gas.) Technically it doesn't change what comes out of your pipes, but they purchase and inject into the system an aggregate amount of biogas equal to what's purchased by customers. So you're directly offsetting burning of NG by instead using the methane that would, best case, be burnt anyway. Of course there is some waste in storage and transportation and such still, but seems like a win. At least, until all the 'free' landfill methane gets used up, and people start creating methane farms.
It's hardly clear that building, installing, and maintaining all the equipment necessary to do that would require less energy than the methane they are wasting.
To a rough approximation, the fact that no one found it financially worthwhile already tells you that you need a ton of resources to make it happen.
Remember to factor in that capturing the methane and burning it is in itself a substantial good in the greenhouse gas equation. Methane is an extremely potent greenhouse gas if allowed to release in its raw form, far more so than CO2 and water vapor - the products of its combustion.
