Besides the comments on your question that already pointed out a few things about Docker best practices you could anyway start a background process from within your start.sh script and keep that start.sh script itself in foreground using the nohup command and the ampersand (&). I did not try it with mongod but something like the following in your start.sh script could work:
#!/bin/sh
...
nohup sh -c mongod --dbpath /test &
...