The closure of a non-rational four-tangle can yield an unknot.
Here is one family of examples. Start with a nontrivial two-tangle (that is, an arc embedded in a three-ball, which is not boundary-parallel). Double it to obtain a "ribbon four-tangle". One closure gives the double of a non-trivial knot. (So the ribbon four-tangle is not rational.) The other closure (capping off parallel ends) yields an unknot.
Here is a picture of the simplest example:
