I think in your example it is in fact irrelevant. I think the main difference is, well how I would use it anyways, is like this:

I would never do something like that

Meaning that you want to tell the other person that you would disagree with him doing what he is about to do, pointing out that you have a "problem" with it.

As opposed to

I would personally never do something like that

In which you want to tell the person that you would never do that, and maybe convince him otherwise, but that you wouldn't mind if he did it anyways.

PS: Don't hate on my spelling/grammar, English isn't my main language but I tend to believe a have a good feeling for it, but feel free to correct me where needed :P

I think in your example it is in fact irrelevant. I think the main difference is, well how I would use it anyways, is like this:

I would never do something like that

meaning that you want to tell the other person that you would disagree with him doing what he is about to do, pointing out that you have a "problem" with it.

As opposed to:

I would personally never do something like that

in which you want to tell the person that you would never do that, and maybe convince him otherwise, but that you wouldn't mind if he did it anyways.

P.S.: Don't hate on my spelling/grammar, English isn't my main language but I tend to believe a have a good feeling for it, but feel free to correct me where needed.