What does “-bot” of “Obama-bot”, “GOP-bot” mean?

That is being used pejoratively and clippingly to compare them to the full word robots.

from the Wikipedia entry linked above:

Fore-clipping

Fore-clipping or aphaeresis retains the final part. Examples: chute (parachute), coon (raccoon), gator (alligator), phone (telephone), pike (turnpike), varsity (university).

Bot is a short form for robot, typically used on the Internet to refer to a program that does something in an automated way. These programs are controlled by some central computer which instructs them what to do. For example, a spam-bot is a program that sends out spam email messages. Spam-bots are installed by viruses and are not under the control of the user of a computer, but are rather remotely controlled by a hacker.

The phrase "ruthlessly programmed GOP-bot" seems to be a direct analogy to the spam-bot. It means a Republican party supporter that does the bidding of the party without any thought of his own.