You can read the file in memory, move the line to where you need it, and write the file back. You can use ReadAllLinesand WriteAllLines.
This code moves the line iup one line:
if (i == 0) return;
string path = "c:\\temp\\myfile.txt";
string[] lines = File.ReadAllLines(path);
if (lines.Length <= i) return;
string tmp = lines[i];
lines[i] = lines[i-1];
lines[i-1] = tmp;
File.WriteAllLines(path, lines);