How to collect 3000 specific file using filename from 2,50,000 files in another directory ?

What you have is O(N*M) where N is the number of names in arrFileLines and M is the total number of files returned by the Directory.GetFiles. This can be changed pretty easily to be O(N+M) ... lots faster.
First, load the arrFileLines entries into a HashSet(Of String):
(My VB is worse than rusty...but here goes.)

Dim matchNames As New HashSet(Of String)(arrFileLines)

This is O(N), it depends only on the length of arrFileLines, and is done only once. (In fact, it could be initialized in whatever loop gets the values into arrFileLines instead of building that array (unless the array has additional uses).)

Then scan through all of the files with Directory.EnumerateFiles once and check for matches. (Use Directory.EnumerateFiles instead of Directory.GetFiles because it doesn't require returning all of the files in a huge array.)

For Each _file As String In Directory.EnumerateFiles(sSourcePath, "*.*", SearchOption.AllDirectories)
  Dim fileName as String = Path.GetFileName(_file)
  If matchNames.Contains(fileName) Then
    File.Copy(_file, Path.Combine(sDestPath, fileName))
  End If
Next

The HashSet allows for checking all of the names in arrFileLines at once in constant time.
This part is O(M), as it depends only on the total number of files returned by the Directory.EnumerateFiles().

Advice: Running your program with a profiler would help you to see where you spend running time.

The profiler would also help you to see if changes are efficient.

When it comes to optimization, there is many tracks to follow to see which one is the best. This were the profiler comes to help you.

Your program is brute force, and some optimizations can be done.

When you got a match and copy the file, you don't need to check the SsourcePath until the end:

For Each _FileName As String In arrFileLines
     For Each _file As String In Directory.GetFiles(sSourcePath, "*.*", SearchOption.AllDirectories)
        If _FileName = Path.GetFileName(_file) Then
             File.Copy(_file, Path.Combine(sDestPath, _FileName))
             Exit for
        End If
     Next
 Next

you also can take advantage of the system

For Each _FileName As String In arrFileLines
     For Each _file As String In Directory.GetFiles(sSourcePath, _FileName, SearchOption.AllDirectories)
        If _FileName = Path.GetFileName(_file) Then
             File.Copy(_file, Path.Combine(sDestPath, _FileName))
        End If
     Next
 Next

Theses 2 tracks are not fully optimized and can be combined. You have to try different solutions ans see how good they are.