I've never burnt willow or poplar. But most of the willow trees round our area (Wensum Valley) have been coppiced (like the Whomping Willow in Harry Potter) I think folk used to make baskets from the thin cut-off branches.
An important thing about burning wood is that it must be seasoned for at least a year after cutting. All our neighbours with wood-burners have several woodsheds, already full of sawn logs.