How do you tag an open air chess board table?

Instead of providing a quick reply like "it does not exist", I will try to show you how finding tags works in OSM.

First, go to the wiki documentation and use the "search" field with "chess". This will show you a list of wiki pages containing the word 'chess'. Usually, only the top 2 or 3 on the list are relevant. Here, the first result is the good one : Tag:sport=chess.

We can see on this page that "the chess board drawn onto a table" is mentionned but is empty (at the time I write this answer). Ouch ! This is not usual in the wiki. It means that this kind of objet is not formally documented on the wiki. But it does not mean that somebody did not tag such thing into the OSM database. To verify this, use the link to 'taginfo', a tool collecting all tags and statistics directly from the OSM database itself. From this link, you will find statistics about "sport=chess" but you can use the 'search' field to check if a key or key/value combination already exist in the database (and how many) about chess tables.

Searching 'chess' gives me one element tagged 'chess=yes'. Then I find 46 'amenity=table' and 12 'leisure=table' but none for "amenity=chess_table" nor "leisure=chess_table". We can conclude that such specific object has never been tagged until now (or with an uncommon wording or foreign language). It is therefore up to you to decide if you create a new one like 'leisure=chess_table' or 'amenity=chess_table' or a combination like 'leisure=table' + 'chess=yes'. You might also contact the dedicated mailing list talking about OSM tagging : tagging@openstreetmap.org and raise the question there. If a consensus is found (or nobody complains about your choice), it would be even better to add your solution back into the wiki documentation if you want that other contributors reuse your tag later.