Can I swap hard drive from my Macbook (10.5.8) into Macbook Pro (10.6)?

No, once you put the older MacBook drive in, if it even fits, you can't transfer to the new one because it's not physically in the MacBookPro anymore.

And the MacBookPro may not be able to read a 10.5 boot disk if it shipped with 10.6.

They are not backwards compatible when trying to boot from an older machines drive that probably doesn't have the correct drivers to run on a MacPro.
First,
Try using Disk Utility to do a Disk Repair, as shown in this link, while booted up on your install disk in case you have some directory corruption.
Let us know what errors Disk Utility reports and if DU was able to repair them. Disk Utility's Disk Repair is not perfect and may not find or repair all directory issues. A stronger utility may be required to finish the job. Better, run DiskWarrior if you have it.
After that Repair Permissions.
No need to report Permissions errors....we all get them.
Then use Target Disk Mode and Migration Assistant to move settings and data.

Make sure all your apps. are 10.6 compatible. I would trash any plug ins also.
I followed this routine and had no troubles.

DALE

You CAN do this maneuver if you have an external enclosure to use temporarily for the extra drive. Or a third external drive you can use for a Superduper bootable backup. Basically you need to make a bootable copy of your current drive on something external. Then install the larger drive in the machine. Boot the machine from the external drive. Then format the now internal drive, and make yet another superduper backup from the external drive, to the larger internal drive.

If you have an enclosure for the new drive, you may be able to save the second superduper copy, but either way will work. I have done this maneuver several times, upgrading to a larger drive, followed by a hand-me-down process of two older machines. Basically I bought one new drive, and upgraded three machines in the process. Takes some time, but easy enough to do.

You cannot simply swap the drives and expect it to work on different hardware. The OS installation is unique for different models of the Mac. But the hardware is generally compatible across the product line, with a new installation.