You mean a wayshirine or portal like in quest, such as a mage's guild quest with Shalidor?
Well... i think i can answer both... none, there is no cool down.
Wayshrines will begin to cost more if you "teleport" to it without using a wayshrine depending on your frequency of use. However using a wayshrine to port to another wayshrine is always free... as is teleporting to another player.
The quest portals dont close... unless of course you completed the quest or is bugged somehow.
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