The largest rainforest in Burundi is contiguous with Nyungwe National Park in Rwanda, and is believed to still be home to hundreds of colobus monkeys. There are also a number of chimpanzees present (although they're very hard to see).
Accsess to the park is via the northern town of Kayanza (which had several places to stay of reasonable standard), but the park itself is fairly uncharted and you would do well to organise a visit through a tour agency.