Why does my chest burn when I cough and nothing comes up?

I presume that you have just had a cold or flu. There can be two main reasons for this.

Firstly, it could be that the lining of the tubes in your chest or upper respiratory tract (trachea and bronchi) are still infected and inflamed. This can be the case even if you are not coughing up phlegm. If so, using Echinacea will help resolve any residue of infection.

An alternative reason could be that you have strained the muscles in your chest with a previous bout of coughing. If this is the case, your chest wall can be tender to touch. Using Arnica Gel externally, rubbed onto the tender parts can help.