Can you use sugar soap to clean a picket fence before painting and will it hurt any plants?

First of all YES sugar soap is harmful to plants and animals including humans so great care when handling as its a form of caustic soda and will cause eye and skin damage very easily.

If the fence is already painted and needs cleaning to remove fungi etc then you can use this product but you can also jet wash the fense and even use a weak bleak solution (thi is also has harmful as sugar soap) - The concept here is removing grease and dirt deposities. The fense can be rubbed down with abrasive wire wool and wasked in strong soap and dried off. It may need an undercoat/promer to make a good key for the top coat.