WARNING: Use this at your own risk
As explained here, from Tainacan version 0.21.5 on, PDF content is previewed inside Tainacan items using the native Broswer PDF reader.
Previously a version of the famous PDF.js library handled this but it was removed due to security reasons. This does have one side effect because mobile browsers and some legacy browsers do not render PDF natively, offering a download button instead. If you really, REALLY need this feature, you may use this plugin at your own risk.
Simply install the release .zip via WordPress Admin or move to your /wp-content/plugins folder and extract there. Make sure Tainacan plugin is already installed and activated then install this plugin itself.