What is the blue film that covers the PCB in my TV's remote?
I have not seen this construction in the wild, but it looks to me like a printed dielectric layer and a printed carbon ink layer overtop of that. That allows a much cheaper punched single-layer paper-based phenolic PCB to be used- the carbon ink acts both as contacts for the silicone keyboard keys and as jumpers. 

Lack of recognition of a button press could be because of a faulty connection between copper and carbon, a dirty contact on the silicone or PCB side, or a cracked PCB or lifted pad (remotes take a lot of abuse and come in intimate contact with greasy human paws and sometimes food and drink). They are also conveniently bone-shaped if you happen to have a large dog.