The problem may or may not be transfered (most likely not transfered). If the problem you're experiencing is some type of virus or other malware if you transfer all the data from one drive to another it would transfer that virus or other malware as well.
However, based off your description I'm assuming the problem you're encountering is either a hardware related issue or software configuration issue. If it's a hardware issue (e.g bad hard drive) the problem would not transfer to the new hard drive, some corrupt files may be transfered though. If it's a configuration issue, depending on what type of configuration issue it is it may or may not be transferred over.
If you're planning on backing up and restoring I'd simply suggest backing up personal data: e-mail, bookmarks, documents, etc. and not ALL the data on the hard drive.