How can I remove my old Apple ID from my iPhone?

You could do this: What to do before you sell, give away, or trade in your iPhone, iPad, or iPod touch https://support.apple.com/en-us/HT201351

Or your could:

Change your Apple ID https://support.apple.com/en-us/HT202667. 

And, here is What to do after you change your Apple ID or 

Password https://support.apple.com/en-us/HT204071 (supplementary to what the fist link says)

Use the article below as a guide of where to sign out of the ID.

What to do after you change your Apple ID or password - Apple Support.