Yes, you can certainly use 'go missing' with objects.
If you say 'The diamond ring went missing', it means that it 'disappeared' ( and the implication is that it was probably stolen). Einstein's brain also went missing.
And just a bit of language trivia - this expression is one of the very few which American English has adopted from British English in fairly recent years. The vast majority of language influence has been in the opposite direction for the last century, but 'go missing' is a rare exception.