As Teapotio is built on top of Symfony2, there is a documentation dedicated to deployment tools.
By now, you probably installed Teapotio on your computer, if not read the documentation about installing Teapotio.
The environment on your server may differ with your local environment, check the documentation about configuring Teapotio.
For your dependencies, do not run
$ composer update on your remote/production server.
Instead, run the command locally, test your code, send the composer.lock file to your server and run
$ composer install on it.
When you run composer install the user running the command may differ from your web server's user. It can result in 500 errors on your server. Check the documentation about setting the permissions.