Self-Destructing Items

If an item tuned to your Echo, other than a Rune item, has the explicit effect of destroying itself after using its effect, this property still applies to the Echo if and when used. In other words, yes, your Echo can and will be destroyed unless your Echo also has the Unbreakable property from its Ruby Tear. In the event your Echo does have the Unbreakable property, the self-destruct effect is still applied, but rather than losing the whole item, the Echo simply loses the tuned material.
Rune items on the other hand are not destroyed when transferring their rune to another item. Instead, the Echo simply loses the rune item’s effects. When the item in question is removed from the Echo, its Frequency is lowered as if the item had never been tuned to the Echo.