We already have a "Connection" for "Discriminatory Entry requirements". This one below isn't exactly "Discriminatory" as it is open to anyone apparently to "do the necessary" if they wish to gain entry!

Ancient Kyoto - Saiho-ji Temple. "Until 1977, the temple was open to admission to the general public, but currently visitors are admitted only by prior application (by return postcard), and the fee to visit the temple (¥3,000) is the highest in Kyoto. Before being permitted access to the garden, visitors must engage in up to two hours of zazen (Zen meditation) and hand-copying or chanting of sutras. It is said that these regulations were put into place in order to protect the delicate moss from the hordes of tourists that plagued the temple prior to 1977."

I wonder if we could find another 2 "Unusual" (but not discriminatory") entry requirements - I thought there might be some along the lines of having to enter and win a lottery to get in but none of the places I can find where that applies is currently a WHS (e.g McNeil River Falls Alaska). I also knew of some Buddhist temples near Mandalay which wouldn't allow "menstruating women" to enter - though how they checked this I don't know. I once had my hair-length checked before being allowed into Hoxha's Albania but haven't had any similar check at a WHS - anyway, you can see there are plenty of opportunities for thinking up some creative controls!

One can't go to visit the Gorillas at Bwindi if one has a cold - so that could be one, albeit perhaps too "sensible" for this purpose!

You have to win a lottery to get into Newgrange passage tomb (Bend of the Boyne, Ireland) on the day when all the action is, namely the winter solstice - 21 Dec. On the shortest day of the year IF there is sunshine, which isn't a given, the rising sun's rays enter the passage and illuminate the tomb area. It must be quite spectacular to behold.

You can enter a draw at the visitors centre there; last year 50 names were chosen from some 30,000+ applicants, which are long odds (and, in the event, the day was overcast which must be a huge disappointment if you get that far).

For the rest, you can of course get in on a guided tour all year round, and they do simulate the experience on these tours.

Thanks FoxPerignon!
So, Els do we have a trivial "Connection"?

Unusual Entry Requirements/Restrictions
WHS which, either permanently or on occasion, require intending visitors to pass "tests" before entry is granted (excluding restrictions on the grounds of race, gender of religion which are covered by their own "Connection")
Bend of the Boyne - New Grange tomb. Entry on the day of the winter solstice is limited to c 50 winners of a lottery (from some 30000+ applicants!).
Bwindi Inpenetrable Forest - Vistors to the Gorillas have to state that they are not suffering from a cold or stomach upset. If they admit before they leave they get their entrance money back but, if discovered along the way, they have to return to base without recompense!
Ancient Kyoto - visitors to Saiho-ji Temple have to undertake a period of Zen meditation and copy or chant Buddhist sutras before entry.

Medieval Monuments in Kosovo: you need to hand in your passport to the soldiers who are protecting the Decani monastery. If they think it's safe, they let you in. I don't know what the requirements really are and how much they "discriminate" because I have seen them letting Albanians get in, so I don't think they refuse certains groups just because of the country or ethnic group you are from. I guess it depends on the soldiers that are on duty that day, combined with the risk of an attack because of recent turmoil.