1. Set mission to have "Respawn on Custom Position".
2. Have only the "Select respawn position" box checked.
3. Set the respawn timer however long you desire. (3 seconds as an example)
4. Play the mission on LAN/Internet, does not matter.
5. Spawn.
6. Assemble the Sleeping Bag/Tent.
7. Disassemble, pick it up, and move to a new location.
8. Assemble once more.

You will see that the old marker/respawn will not go away, and no new marker/respawn is created.

As an additional note, shutting down and then resuming the multiplayer mission seems to "fix" the issue, but creates an entirely different issue. After shutting down the mission and then resuming the mission; disassembling, then assembling at a new location will then create a new marker/respawn. However, the old marker/respawn still does not go away, which then means you now have two respawn locations when there should only be the new one.

UPDATE: I may have some information that can help with fixing this issue. It seems that when you disassemble the Sleeping Bag/Tent, the Bag/Tent entity still remains on the ground despite being disassembled and equipped.