A prefix-dropping iterator must preserve the first satisfying element. Reading it to test the predicate physically consumes it from the source, so keep it in a one-element buffer. Once a match is found, return the entire suffix without testing subsequent elements again.
trait Iter[T] {
  def hasNext: Boolean
  def next(): T
  def copy(): Iter[T]
}

class DropUntil[T](p: T => Boolean, iter: Iter[T]) extends Iter[T] {
  private var seeking = true
  private var saved: Option[T] = None

  private def prepare(): Unit = {
    if (seeking) {
      while (saved.isEmpty && iter.hasNext) {
        val x = iter.next()
        if (p(x)) saved = Some(x)
      }
      seeking = false
    }
  }

  def hasNext: Boolean = {
    prepare()
    saved.nonEmpty || iter.hasNext
  }

  def next(): T = {
    prepare()
    saved match {
      case Some(x) => saved = None; x
      case None =>
        if (!iter.hasNext) throw new NoSuchElementException("next")
        iter.next()
    }
  }

  def copy(): Iter[T] = {
    val result = new DropUntil[T](p, iter.copy())
    result.seeking = seeking
    result.saved = saved
    result
  }
}
Assume the source iterator is owned by the wrapper, its copy meets the stated independent-cursor contract, and is a pure function. Before preparation, the logical remainder is the source remainder with its maximal initial failing prefix removed. After preparation, it is the buffered match followed by the source remainder, or simply the source remainder when the buffer is empty. This describes the abstraction function and proves that repeated hasNext calls neither lose the match nor skip more values.
Copying preserves the phase flag and optional buffered value and independently copies the source iterator. It does not rerun the predicate on an already accepted match. A shallow copy of the wrapper alone would give shared iterator consumption, violating the specification. Therefore
When no element satisfies , a finite source yields an empty iterator. Seeking can diverge on an infinite source with no match; this is inherent in determining whether its required suffix begins anywhere. An element object is not deep-copied by this cursor operation.
The first matching entry must be retained, and copying an already-started prefix-dropping iterator must not restart the search. Use a private initial buffer together with the underlying iterator snapshot:
class DropUntil[T] private (
    p: T => Boolean, iter: Iter[T],
    saved: Option[T], alreadyStarted: Boolean) extends Iter[T] {

  def this(p: T => Boolean, iter: Iter[T]) =
    this(p, iter, None, false)

  private var pending: Option[T] = saved
  private var started: Boolean = alreadyStarted

  private def prepare(): Unit = {
    if (!started) {
      var found = false
      while (!found && iter.hasNext) {
        val x = iter.next()
        if (p(x)) {
          pending = Some(x)
          found = true
        }
      }
      started = true
    }
  }

  def hasNext: Boolean = {
    prepare()
    pending.nonEmpty || iter.hasNext
  }

  def next(): T = {
    prepare()
    pending match {
      case Some(x) => pending = None; x
      case None =>
        if (!iter.hasNext) throw new NoSuchElementException("DropUntil.next")
        iter.next()
    }
  }

  def copy(): Iter[T] =
    new DropUntil[T](p, iter.copy(), pending, started)
}
Before preparation, the iterator still represents the original remaining sequence. During the search, the loop invariant says that every discarded entry fails the predicate, and pending is either empty or contains the first satisfying entry just consumed. For a finite input, the number of unread entries is a loop variant. After preparation, the abstract remaining sequence is pending.toList followed by the underlying iterator's remaining sequence. Repeated hasNext calls do not discard or test more entries after the first match.
The iterator snapshot copies both the cached entry and the started flag. In particular, copying after the matching entry has been returned continues with the remaining suffix, including later entries that fail p. Restarting new DropUntil(p, iter.copy()) at that point would wrongly drop another prefix.
As usual, assume a deterministic predicate without shared mutable effects. On an infinite input having no match, the first demand cannot terminate: no finite inspection establishes absence. Independence here concerns cursor state; arbitrary mutation of shared element objects is outside the iterator contract.
Keep a buffered first match and a flag recording whether the initial prefix still needs to be skipped:
class DropUntil[T] private (
    p: T => Boolean,
    iter: Iter[T],
    private var cached: Option[T],
    private var seeking: Boolean) extends Iter[T] {

  def this(p: T => Boolean, iter: Iter[T]) =
    this(p, iter, None, true)

  private def prepare(): Unit = {
    if (seeking) {
      while (cached.isEmpty && iter.hasNext) {
        val x = iter.next()
        if (p(x)) cached = Some(x)
      }
      seeking = false
    }
  }

  def hasNext: Boolean = {
    prepare()
    cached.nonEmpty || iter.hasNext
  }

  def next(): T = {
    prepare()
    cached match {
      case Some(x) => cached = None; x
      case None =>
        if (!iter.hasNext) throw new NoSuchElementException("DropUntil.next")
        iter.next()
    }
  }

  def copy(): Iter[T] = {
    prepare()
    new DropUntil[T](p, iter.copy(), cached, false)
  }
}
While seeking, prepare drops exactly the rejected prefix and caches the first accepted value, including a possible null payload as Some(null). Afterwards the abstract remainder is the cache, if any, followed by the underlying remainder; later values are returned even when they fail p. Repeated hasNext calls do not lose the cached value.
The iterator snapshot copies the underlying cursor and the cached value, then marks the initial search complete. It does not reconstruct a fresh two-argument DropUntil, which could incorrectly discard later values. Resolving the initial search before copying also avoids having two snapshots independently rerun a privately stateful predicate. The wrapper owns consumption of the supplied source cursor. If that source is infinite and never produces an accepted value, seeking cannot finish. This is a prefix-dropping iterator, rather than a repeatedly demand-driven filtering iterator.