An empty string should work. Not sure how you called the API but try this in Node JS

var mobilePhone = { contact: { mobilePhone: ""}}
platform.put('/account/~/extension/[extensionId]', mobilePhone)
    .then(function (resp)
       console.log(resp.json())
    })
    .catch(function(e){
       console.log(e.message)
    });