The most efficient way to trim an array is to create a new array of the desired size, then use Array.Copy to copy the desired portion of the old array.
I would recommend that you maintain a List<string> containing all of your lines.
You should use a StringBuilder to build a string containing the lines you're looking for, and set the textbox's Text proeprty to the StringBuilder's string. For added performance, set the StringBuilder's capacity to a reasonable guess of the final sie of the string. (Or to list.Skip(...).Take(...).Sum(s => s.Length))
If you're concerned about memory, you can trim the List<string> by calling RemoveRange.
As long as you don't put too much in the textbox at once, doing it this way should be extremely fast. All of the manipulation of the List<string> and the StringBuilder can be done in a background thread, and you can pass the completed string to the UI thread.
The TextBox.Lines property simply concatenates the array you give it using a StringBuilder, so there's no point in using it (and making a needless array).