The body of the iterator can be exactly the same as the one in the sequence:

val fibonacci = buildIterator {
    yield(1L)
    var current = 1L
    var next = 1L
    while (true) {
        yield(next)
        val tmpNext = current + next
        current = next
        next = tmpNext
    }
}

So now we can easily, and on demand, get up to 92 numbers in the sequence from this iterator:

for (i in 0..91) {
    println("$i is ${fibonacci.next()}")
}

If you are curious, these are the last few elements that can be calculated with this implementation: