Apparently Ethiopia is the birthplace of coffee. In the southern highlands of Coffee got cultivated first. Is the coffee plant still growing in the wild in that area, and is possible to go for a coffee picking, burning and subsequent drinking-trip to Ethiopia?
As mentioned in Choster's comment, there are some wild coffee plants in Ethiopia. You can go and pick coffee cherries, for sure.
However, that is not the case for actually drinking the coffee you picked. It needs to be processed. Processing coffee has few different methods; the one that is mostly used in Ethiopia is called the "dry process." This method can take up to 4 weeks to make the coffee beans ready to be consumed.
the other main method is the wet process, which requires special equipment and lots of water, which won't be available in the wild.
The bottom line is that if you are willing to spend few days hunting coffee plants and picking cherries and then waiting from several days to several weeks to get the coffee processed, then the answer is: yes, you can do that. In reality though, I do not think this is feasible.
Not impossible but very hard...
Coffee grows in the shadow and more than 90% of the Ethiopian forest is gone. I am not saying it's impossible, but as you can imagine, this "per se", reduces your chances of finding it in the wild a lot.
Wild coffee does actually exist. I am aware of, at least, some forests, at 180Km south Addis Ababa where you can find it. It's cropped and sold at, of course, much higher prices. If you can find it, I expect that people protect it hard. Ethiopia is a very poor country and such a rare commodity has an high impact in the local economy.
If you come across some wild coffee, before starting to crop it I would advise you to ask first to the locals if doing so is OK. You should maybe even buy it from them. By cropping it you may be depriving them from income and you may also be cropping it before it is ready. You will end up with nothing and so will the local population.