This road here: Way: Truppenübungsplatz Durchfahrt (282653281) | OpenStreetMap is only open during specific days and/or daytimes. You need to check the opening hours. How can I make the routing engines pop-up a infobox, which tells the driver to check if the road is open?
It should look like the pop-ups in routing engines that show a road is unpaved.
You have to ask each routing engine individually if they already support this and then all renderers for the routing-engine if they want to include a popup to override this.
I can imagen like OSMAnd is doing when the destination is in a private street.
Okay, thanks for the information 
This is well-discussed special case, discussed over years within a Note Note: 1152142 | OpenStreetMap which is also referenced by the note-tag of the street. Please accept the result of this intensive discussion, or open a new thread regarding this special case in the Austrian sub-forum, thanks!
(the Note does not only affect the 2 limiting gates, but also the road between them within the military area)
Ultimately, it is up to the respective application how the available information is used and presented. If you want to see a popup in an application, you have to contact the corresponding developer, but it has nothing directly to do with the data. At least some of the conditional restrictions are used by at least a few routers in different ways, either by only using / not using the sections at certain times or by indicating a possible block (arbitrary example which is currently tagged with access=no + access:conditional=yes @ 08:00-17:00)