You are correct that a gerund is necessary after "Don't mind me...". You can't say "Don't mind me *to change the channel".
I should point out that the meaning of the two phrases is very different though!
You can't make a replacement of one phrase for the other because "Would you mind if I..." is asking for permission, whereas "Don't mind me..." is simply informing someone that you are doing something. The latter phrase is way more limited in how you could use it.
"Don't mind me changing the channel" is something an English speaker would say if the other person were studying and not watching TV, for instance, and you were warning them the sound on the TV may change abruptly, or if you were reaching over them to grab the remote control.
You aren't asking for permission, but simply informing them that you are doing it.
The link you gave implies something different to what you initially thought; saying "Would you mind..." and "Do you mind..." are very similar ways to ask for permission. "Don't mind me..." is not in this category.
Hope that helps!