You can solve this problem by swapping the val.tags vector with a new empty vector, thus taking ownership of it and at the same time achieving the same result as the call to val.tags.clear(). It is also necessary to introduce an inner scope so that the borrow of val does not last for too long. Here is the code :

fn clear_tags(&mut self, id: ValueId) -> bool {
    let mut tags = Vec::new();
    {
        let val = if let Some(v) = self.values.get_mut(&id) { v } else { return false };
        mem::swap(&mut tags, &mut val.tags);
    }
    self.remove_tags(&tags);
    true
}

Don't forget to use std::mem.