AFAIK, there is no a interactive option for output to file, there is a previous SO question related with this: Printing mongodb shell output to File
However, you can log all the shell session if you invoked the shell with tee command:
$ mongo | tee file.txt
MongoDB shell version: 2.4.2
connecting to: test
> printjson({this: 'is a test'})
{ "this" : "is a test" }
> printjson({this: 'is another test'})
{ "this" : "is another test" }
> exit
bye
Then you'll get a file with this content:
MongoDB shell version: 2.4.2
connecting to: test
> printjson({this: 'is a test'})
{ "this" : "is a test" }
> printjson({this: 'is another test'})
{ "this" : "is another test" }
> exit
bye
To remove all the commands and keep only the json output, you can use a command similar to:
tail -n +3 file.txt | egrep -v "^>|^bye" > output.json
Then you'll get:
{ "this" : "is a test" }
{ "this" : "is another test" }