The "using" statement around your MemoryStream disposes it when execution reaches the closing brace } at the end of the using block. This is precisely the reason why you write using, so that it frees the resources used by the MemoryStream. Therefore, if you do not want to release it, but rather you want to keep it in memory, you should not be writing the using statement. Instead, declare the MemoryStream variable wherever you want to keep it, for example, at Form level. And then, when you do not need it any more, call memoryStream.Dispose() to free it.
However, in this particular case, doing all these things is overkill. You can load the whole file into an array of strings using File.ReadAllLines, as mentioned in a previous answer. This is much shorter to write than the memorystream plus the streamreader, and you can easily keep the array of strings in memory as long as you wish. It does not need to be Disposed.