The exif info, i.e. the timestamp created by modern digital camera, will be preserved upon export should you choose to do so.
I don't know what you mean by "on the file." Are you talking about a watermark?
Also, note that the file date and timestamps may be different than exif date and timestamps. Both on your original and on the exported image. It's important to know which ones you're referring to. Look in the metadata panels in Lr and you'll see different date and timestamps on occasion.
If you mean that you want the file creation date to match the image capture date, no, there is no way to do that in LR.
I am sure there are other utilities that can do that for you, but why?
Any properly written picture viewing app will be looking at the EXIF data to read the capture date, never the file date.