To get your podcast listed on TuneIn, you'll need to head to this link: https://tunein.com/podcasters/

Then, click on Publish Podcast out of the two options. This will take you through to the Submissions page. The form will look like this:

Simply fill in all your details, you don't necessarily have to enter things like your phone number and Twitter handle if you don't want to! Once you've filled in everything, just click Submit. TuneIn will send you an email confirmation when your show is up in its directory.
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​Note: Station Email doesn't have to be filled in. This is likely left over from the radio stations submission form that no longer exists on TuneIn.