I am going to replace a hard drive that has 204 reallocated sectors, but if I use Clonezilla to duplicate my files, settings, and programs to the new hard drive, will these sectors affect my new drive in any way?
The reallocated sectors are drive specific and do not mirage to a new drive. It is simply the drive pointing out to itself, "Hey, I can write data every where but here, here and here" (204 "heres" in your drives case). When the disk identified that the sector is troubled, it moves any data in that sector to a safe place and remaps it's activity to work around that sector. This is why when the reallocated sector count rises, disk performance can show a noticed degradation.
If all your files, settings and programs are working fine now, they will be fine after the cloning.