City of Austin will pick them up on those large item trash days they have about every 6 months in neighborhoods where they pick up refrigerators, couchs and stuff. You are supposed to drain the gas and oil.
Or a shop that repairs lawn mowers might take it for their lawnmower junkyard. Some places have these.
Or, what's that place downtown on 8th or 9th and IH-35 that takes all the recycling items? They would either take it or know the proper place to take it. Is it Ecology Action?
This is not politcally correct, I know, but there is an open dumpster behind the grocery store next to my house. I throw crap like that in it all the time late at night.
Likely depends on you neighborhood, but my brother justs puts thing like this in front of his house with a For Sale on it. They generally get hauled off during the night.