Any fabrics that block visible light without blocking IR/Bluetooth?
Unless you get metal lined material or mirrored glass doors the radio won't have an issue. And to be honest even if you did it would probably still work fine by going through the wood or whatever the sides of the shelves are made of unless they are solid metal.

Any solid fabric is going to block the IR. You could potentially go with a fairly lose weave fabric so that the light can get through the holes. The remote will probably still work but you will also still be able to see the blinking lights.
If the amp is right at the front of the shelf you could potentially put a small hole in the curtain in line with the IR receiver in the amp to allow the remote to get through. But if there is any separation between the two you end up either needing a big hole or the remote only works from one direction.

A dark tinted glass or plastic may be your best option. Some will block IR, some won't. The problem is being able to test the material without having to first buy some doors made out of it.
You could do a test, get some of the stick on limo tint plastic, it's fairly cheap, and see if your IR remote works through that with sufficient range. If so then get some plain glass doors and add the tint. That wouldn't completely block the blinking lights but would make them a lot dimmer.

The other option is the more direct approach, put something over the blinking lights. You could use something solid like a sheet of black plastic in front of everything with a cutout where the IR receiver is.  Or you could use a dark colored tape and put it only over the lights.
Visually not always the nicest approach but once everything in there is dark then you could put some fairly transparent glass doors over the front and it wouldn't be visible.