We have a similar situation. We are in the country with no trash pick up. We recycle, compost, burn, and make 3-4 dump runs a year with our own garbage. Our first guests generated more trash in 5 days than we create in a year. We had to hire someone to make a big dump run, which cost us more than our cleaning fee.
I've created new house rules regarding trash, asking guests to seperate their recyclables and what can be burned from their trash, and asking them to take their trash over two 40 gallon bags with them when they leave. I don't know how this will work out, but I have stayed in one other airbnb before I started hosting that required us to take our garbage with us, so I know it's not unheard of.
We are new hosts; I'll let you know how our new rules work out - we have the same guests coming to stay in a couple of weeks.