Hello, Not necessarily that you should have a smart TV. There are many methods/apps which can help in casting video from your Phone to Tv through DroidBOX. Check this video and see if that helps. In the video i have shown how can you cast videos from Youtube to your TV screen. Check if you can do in a similar fashion with your phone Chrome browser.
If you have content stored on your phone you can use the Android Es File Explorer app to share the content on the same network and then access the same files through your DroidBOX and play them on your TV.