If LPub is configured to use LDView as its renderer, and LDView is configured to automatically download unofficial parts, then the file should load fine on the new computer as long as all of the unofficial parts you use are on the Parts Tracker.
However, if LPub is configured to use POV-Ray as its renderer, then I believe it will only work if L3P can find the parts, and L3P doesn't automatically download unofficial parts. (This presupposes that LPub uses L3P to generate POV-Ray output, but I'm pretty sure it does.) If you open the file in LDView, LDView will download the unofficial files needed for you automatically, then L3P should be useable.
Having said all that, L3P (1.4 Beta) may or may not automatically support the Unofficial directory that LDView downloads parts into. I think it does, but I could be wrong about that. If it doesn't, you should be able to still get it to work by creating a properly formatted ldraw.ini file, but I'm not going to go into details about that unless it doesn't work automatically. So, if you load the file in LDView on the computer and all the parts display, but then some are missing when rendered in LPub using POV-Ray, let me know, and I'll attempt to provide instructions for an ldraw.ini file with Unofficial directory support.
However, if LPub is configured to use POV-Ray as its renderer, then I believe it will only work if L3P can find the parts, and L3P doesn't automatically download unofficial parts. (This presupposes that LPub uses L3P to generate POV-Ray output, but I'm pretty sure it does.) If you open the file in LDView, LDView will download the unofficial files needed for you automatically, then L3P should be useable.
Having said all that, L3P (1.4 Beta) may or may not automatically support the Unofficial directory that LDView downloads parts into. I think it does, but I could be wrong about that. If it doesn't, you should be able to still get it to work by creating a properly formatted ldraw.ini file, but I'm not going to go into details about that unless it doesn't work automatically. So, if you load the file in LDView on the computer and all the parts display, but then some are missing when rendered in LPub using POV-Ray, let me know, and I'll attempt to provide instructions for an ldraw.ini file with Unofficial directory support.