I'm trying to select the most recent file to import into my table with the most recent date
What type of file? Text, CSV, Excel,....???
There will be other files in the folder.
What types of files? Same file type? Different file types? You might need another "IF...END IF" to check for the file extension before you check the file date.
You must be more specific with your question.
I am confused. You use the FSO to search the folder "\\10.3.0.144\RSH_Log\DATA", but you are importing from a different folder in the DoCmd.TransferText command: "\\10.3.0.144\RSH_Log\DATA\4300R TEST DATA PLOTS\import.txt"
Usually you import the file from the same folder you search in. 
If the folder has ONLY text files and you want to import the latest file, concatenate the file name to the path:
Code:
Public Sub ImportFile()
Dim objFile As File
Dim objFolder As Folder
Dim objFSO As FileSystemObject
Dim FolderToScan As String
Dim NewestFile As String
Dim NewestDate As Date
Set objFSO = CreateObject("Scripting.FileSystemObject")
FolderToScan = "\\10.3.0.144\RSH_Log\DATA"
Set objFolder = objFSO.GetFolder(FolderToScan)
NewestFile = ""
NewestDate = #1/1/1970#
For Each objFile In objFolder.Files
If objFile.DateLastModified > NewestDate Then
NewestDate = objFile.DateLastModified
NewestFile = objFile.Name
End If
Next
CurrentDb.Execute "DELETE * FROM tImport", dbFailOnError
'--------------------------------------------
' testing - is the path and file name correct???
MsgBox "The path and file name to be imported is " & vbNewLine & vbNewLine & FolderToScan & "\" & NewestFile
' Delete or comment out when testing is complete
'--------------------------------------------
' DoCmd.TransferText acImportDelim, "tImport Specification", "tImport", "\\10.3.0.144\RSH_Log\DATA\4300R TEST DATA PLOTS\import.txt", False
DoCmd.TransferText acImportDelim, "tImport Specification", "tImport", FolderToScan & "\" & NewestFile, False
MsgBox "Data has been imported into the tImport table"
Set objFSO = Nothing
End Sub