Having a pasture is not essential, but your donkey will be happier to have some place to run and graze and play. Depending on your land and the size of your animals, you can figure on pasturing one or more donkeys per acre. Turn out on green pasture should be supervised. Donkeys will founder on rich feed.  Be prepared for a dust crater somewhere, however. Donkeys love to take dust baths and will pick a spot in the pasture to dig out and 'bathe' themselves daily.  

Try to control grazing and don't allow your donkeys to graze their home into a dust bowl. Most donkeys shouldn't be left year round on pasture anyway, they get too fat and could possibly grass founder, so rotate fields or allow your donkey only so many grazing hours per day.   If possible,  rotate your pasture use, this will help keep the grass healthy and cut the parasite cycle. Harrowing your field can also be beneficial.  Try and do this in the hot part of summer, when the sun will kill any lingering parasites in the fecal piles as they are broken open and spread around from the harrow. Managing the land along with your donkeys helps you control weeds, parasites and soil erosion, and then you benefit by having a pretty, healthy place for your donkey(s) to play in.  

Donkeys are not normally escape artists, unless opportunity arises. Make sure you have secure latches on your gates. A four foot fence will be adequate for most occasions. You might consider having woven wire (non climb fencing) around the bottom part of the fence to protect the neighboring dogs from your donkeys!

Donkeys are usually smart enough to not get caught up in wire, but to be on the safe side make sure all fencing is tight and secure. If you can't get away from having barbed wire (it's not recommended, considered the worst fencing for any equine), at least use insulator extensions and run an electric 'hot' wire around on the inside that will help keep the donkeys away and scar free.

Protect all wood surfaces, with either metal flashing covering the exposed corners or with an anti chew paint. Some donkeys can make even a beaver envious. 

Plants Toxic to Animals:
http://www.library.uiuc.edu/vex/toxic/toxic.htm
or
http://www.ansci.cornell.edu/plants/index.html