My feedburner podcast feed won't publish in Itunes

Your feed is not a podcast feed. It has no 'enclosure' tags containing the URLs of your media files, and indeed appears to have no reference to media files in it anywhere.

it's not clear how you are creating your feed, whether in Feedburner itself or in another program or service before submitting to Feedburner. Looking at the coding for the top episode there is a link to a web page for that episode containing an embedded player: this sort of thing won't work in iTunes. You must provide a direct link to the media file - in the case of that episode it looks as if it may be

ia800308.us.archive.org/11/items/OldFashionedMarketingTheModernWayWithScottForre st201505141830/Old-fashioned%20marketing%20the%20modern%20way%20with%20scott%20f orrest%20-%202015-05-14%201830.mp3

preceded by https:// (the forum truncates the URL if I post it in full). You should be using http not https.

Incidentally the quality of this file is terrible, with serious compression artefacts and even cutting out meaning that the speech is actually unintelligible in places - your bitrate of 122kb/s stereo is a bit low but I wouldn't have expected the results to be that bad. You may want to try using a higher bitrate (less compression); are you sourcing some of the speakers via Skype or another VOIP service? that might cause problems.

You may find it helpful to look at my 'Get you started' page on podcasting: http://wilmut.uk/pc

The file URL I posted no longer works and the player has disappeared from the website page for the episode. When I listened to a few short bits of the file I though the quality was pretty poor, but if you are happy with it, it's your podcast and your decision. I just offered it as an an additional comment.

In the meantime, you don't submit the mp3 file URL to iTunes, you have to place within the 'enclosure' tag in your feed - either by getting it into the original feed or using Feedburner's 'Smartcats' facility.

When the feed is working, and has been accepted by the iTunes Store, all you have to do is to add episodes and re-publish the feed; the Store will update automatically after a couple of days or so and subscribers will see the change immediately.

Also, from what you say there may be a little confusion over teminology: a podcast is the overall 'wrapper' for a number of episodes; you submit the 'feed' which contains all the information to iTunes, and then the episodes will be listed and can be updated. You can't place individual episodes in iTunes, only the feed with the episodes contained therein.

People do use archive.org to host their media files, you just have to obtain the actual file URL by looking at the list of alternative media on the right, and copying the URL for direct access to the media file. You need to check it in a browser and make sure that it plays there, not downloads or leads to a web page or an embedded player, and the URL should begin with http not https (just removing the 's' should work). Otherwise you can host media files on any ordinary web server provided that it supports 'byte range requests' (the method used by iPhones to access the file, and mandatory in iTunes). If you provide me with the URL of a media file I can check for this.

You then have to get that into your feed. As the feed doesn't indicate what the method is you are using to create it I can't say what you should do to achieve that. The point is that it has to be a podcast feed, not just a blog feed - the media file URL has to be within an 'enclosure' tag, not just a link in a description or an embedded player.

You can test your feed by subscribing manually in iTunes from the 'File' menu. If the episodes with media files appear then you are on the way to success. If they don't, then obviously you have more to do. Episode which don't have properly referenced media files won't appear in iTunes.

It's no good using the 'embed' code. The feed must have the actual MP3 file URL which is what you give, but you should change the http to https. (The server does accept byte range requests, so that's OK.)

As you are using WordPress, you may need to use a plugin such as PowerPress - I'm not familiar with WordPress but I get the impression that it won't create podcast feeds on its own. Certainly the feed URL you post is not a podcast feed - unlike the version I looked at earlier it doesn't include the 'iTunes declaration' in the top line, nor any 'itunes:...' tags - without an 'itunes:image' tag containing the URL of an image which is at least 1400 x 1400 px (and less than 500kB) your feed won't be accepted even if it contains the media files (which it doesn't). Again, links in description tags or embedded players will not work in iTunes.