Can I upgrade the logic board of an iMac (late 2009) to a later model?

Sadly Apple made large changes between the series as such you won't be able to swap out your logic board.

There are a few things you can do to improve performance like upping the RAM (8 GB or 10 GB)

Upgrading the HD or better yet setup a dual drive using a SSD drive.

Here's the IFIXIT guide and part you'll need for a dual drive config: Installing iMac Intel 27" EMC 2309 and 2374 Dual Hard Drive. I would recommend a 128 GB or 256 GB SSD here. You'll need to be careful on what SSD you go with as your system only has a 3.0 Gb/s SATA port. So you'll need a drive that supports this slower speed.