So I recently made the switch to cycles rendering and smoke modifiers, but it's been giving me loads of problems already.
This is what the model looks like: 
And then when I go to render it, this is the result I get.
Why is this happening?
So I recently made the switch to cycles rendering and smoke modifiers, but it's been giving me loads of problems already.
This is what the model looks like: 
And then when I go to render it, this is the result I get.
Why is this happening?
It looks like you have a diffuse shader assigned to the surface of your domain. Make sure that there is nothing (or a pure white transparent shader) plugged into the surface socket on your domain object, and that the smoke nodes are plugged into the volume socket.

Problem AMD video cards do not support Smoke or fire rendering, instead of the fire or smoke you will get the domain rendered as a solid object.
Solution: Switch to CPU only on system preferences, or get an Nvidia video card.
Sadly there is no word on AMD support for Blender Fire and Smoke rendering.
There is a way to disable the emitter for particles under the Particle properties tab in viewport display and render (Uncheck show emitter).
For (fog boxes) you should only need to go to the viewport display option in the same tab and set display as Wire or bounds.
Don't know if this works for fire & smoke.
VERY SIMPLE METHOD: Highlight the domain bounding box, then in the scene menu, top right, turn off the little camera icon and it will no longer appear in your render.