Node.JS Server Queue Processor
The structure looks fine. It is quite succinct and makes good use of const and let where appropriate. 

To adhere to the D.R.Y. principle, process() can and should utilize remove() to remove items from the queue.



I considered suggesting that arguments be accepted with each function but that can be achieved with partially bound functions. 

I also considered suggesting you consider using a class, since ES-6 featured can be utilized, but then you would either need to instantiate a queue once or else make all methods static. 



I would suggest you consider accepting an error handler callback for each function. That way, instead of writing all errors to the console, the caller could add an appropriate handler.