How to change Apple ID to manage podcast?

If you want to move a current listing for a show in Apple Podcasts to a new ID, contact support by clicking the Contact Us link at the bottom of this page:

https://itunespartner.apple.com/en/podcasts/overview

Tell them you want to move the listing to a new ID and they'll give you instructions on how to do it.

You say you 'lost the old ID' - I assume you mean you don't have the email address which forms your ID. You can change this as detailed below:

If your primary address (the ID) is an @icloud.com, @me.com or @mac.com address you cannot change it. If it is a non-Apple address you can change it as follows:

1. Firstly, sign out of iCloud on all your devices: go to each of your devices and sign out in System Preferences (or Settings)>iCloud - 'Sign out' on a Mac, 'Delete this account' on an iOS device (this will not delete the account from the server). If you have 'Find My iPhone' etc enabled you will need to disable it first. Your synced data will disappear from your devices but will reappear when you sign back in. Also sign out in your account in iTunes.
2. Create a new email address, for example at Yahoo or Gmail, or anywhere convenient (or you can use an existing address as long as it is not associated with an Apple ID). You can change it to another Apple address only if it's listed when you click the link to change (in practice this means your main @icloud.com address, and the equivalent @me.com and @mac.com versions if you have them: it does not include email aliases).
3. Go to http://appleid.apple.com and click 'Manage your Apple ID'. Sign in with the current ID.
4. Where it says 'Apple ID' and gives your current ID email address, click 'Change your Apple ID'.
5. Enter your new address and click 'Save changes'
6. Sign back into iCloud on all your devices with the new ID and password. Your synced data will reappear (give it time).

I reiterate - before you start, sign out of iCloud on all your devices. Failure to do this will lead you into difficulties.

When you sign back into iTunes with the new ID you should be able to manage your podcast: you haven't actually changed the ID itself, only its name.