Silver Coins In A Birch Box

I am sure the wood would be fine for the coins. You can expect subtle toning over the years, but likely attractive. To slow the toning process, you could seal the wood with some kind of polyurethane. Correct me if there are any wood-toning experts out there.

The greatest problem to a wood box is the friction aka "cabinet rub" that scrapes between the high points of the coin and the wood surface. Also the contact marks between the coins can be an issue.

Eco friendly wooden cutlery is made from birch wood, so I guess it would also be OK for a cabinet or a box made from birch wood to be OK for the storage of coins.

The British Museum uses wooden cabinets for the storage of the collection. The coins themselves get minimal handling , because the most of the collection is photographed and supported with a professionally maintained digital database.

Perhaps you may wish to emulate the British Museum example in some lesser way, but most owners of much smaller private collections wish to handle their coins in some sort of protected way. That is one reason why slabs are used, although I am not keen on them myself.

Another commonly used approach to the storage and display of coins is what I use: that is, cardboard Mylar 2x2's in archival quality 20 pocket coin album pages.

The problem with slabs or 2x2's for the OP is then that the birch wood box is no longer used for coins. Perhaps the box can be re used for some other purpose.
If the collection is very small, 2x2's can still be kept in the birch wood box.

Cedar can influence toning--at least with copper coins over a long period of time.

Aussie Red Cedar is somewhat different. From a carpentry point of view, it is easily worked and has most of the working qualities of a softwood, and takes a high French polish, but it also has some other endearing qualities. Australian Red Cedar is an excellent cabinet timber but not OK for coins due to aggressive resin problems. The resins from this species of timber definitely will the toning of coins.

In fact, Australian Red Cedar used to be used for fence post in the early Australian Colonial days because after surface charring in fire, was almost completely fire proof, as well as termite proof. The reason for this is due to the resins in the wood which termites avoid and also enhances fire resistance after surface charring. Aussie Red Cedar also was an excellent structural timber and is still found in the roof beams of large public buildings of early heritage importance.

Due to it's excellent properties in early Australian Colonial days it was very common then, but is now one of the rarest and most valuable cabinet timbers in the World, with virtually no trees left in the wild.

Australian Pink Cedar is much easier to find, but is also getting becoming much scarcer. Comes from a different species of tree.