It turns out that Redis isn't exactly supported out of the box when you're using Lumen, and the docs don't reflect that. Without the additional dependencies installed, you're going to see this error when you try and use the cache.
Fatal error: Class 'Illuminate\Redis\RedisServiceProvider' not found in /home/vagrant/lumen/vendor/laravel/lumen-framework/src/Application on line 219
There are in fact an additional dependency in order to get Redis up and running.
You need to include
illuminate/redis (~5.0) which includes Laravel's Redis drivers.
composer require illuminate/redis
Once you've installed this dependency you're good to use Redis on Lumen.
I've made a PR for the Lumen docs to illuminate this dependency.