How can I get an extension method to change the original object?

You have to modify the contents of the List<string> passed to the extension method, not the variable that holds the reference to the list:

public static void ForceSpaceGroupsToBeTabs(this List<string> lines)
{
    string spaceGroup = new String('.', 4);
    for (int i = 0; i < lines.Count; i++)
    {
        lines[i] = lines[i].Replace(spaceGroup, "T");
    }
}

You'd have to change the contents of the original list - just reassigning the parameter to have a different value isn't going to do it. Something like this:

public static void ForceSpaceGroupsToBeTabs(this List<string> lines)
{
    string spaceGroup = new String('.', 4);
    for (int i = 0; i < lines.Count; i++)
    {
        lines[i] = lines[i].Replace(spaceGroup, "T");
    }
}

It's worth noting that this has nothing to do with extension methods. Imagine you'd just called:

Helpers.ForceSpaceGroupsToBeTabs(lines);

... because that's what the code is effectively translated into. There's nothing special about the fact that it's an extension method; if you change the code so that the "normal" static method will work, then it'll work as an extension method too. As noted in the comments, one thing you can't do with an extension method is make the first parameter a ref parameter.

(EDIT: I realise this is the exact same code that dtb posted, although we arrived there independently. I'm keeping this answer anyway, as it's got more than code.)

If it's a reference type, you'd have to change it's contents. If it's a value type you're passing in, you're out of luck. The very existence of extension methods is put into place to support functional paradigms in C#, and those functional paradigms, by their very essence, tend towards immutability of types, hence the inability to change the value off of which the extension method is called.

In other words, while you could do it, it may not be in keeping with the "spirit" of functional programming.