I can vouch for this behavior as well.
If I play the game without the bot, I can play for hours on my home PC running a Nvidia GTX 980 Ti, or on my Surface Pro 4 (or any other device really), and see no lag in the game at all. However, when I run the bot the game will start to slow down and show graphical stuttering after a while.
Sometimes this stuttering can take a few games to show up, up to an hour or so. More often it happens during the first game.
When the stuttering starts, misplays begin to happen. A card will be chosen and drug out to the play field, but the graphics will stutter and the card will reset. The bot will continue to try to play the rest of its string and fail because the card isn't in play. So it will recalculate, try again, and maybe be successful. Eventually, this results in enough of a stutter that the bot is constantly recalculating the play field and starts brushing up against the timer, sooner or later not making the timer anymore.
I've noticed that if I play a Pirate Warrior deck, this problem takes a longer time to show up. I think it's because that kind of deck never has more than 4 or 5 minions on board at any time, cutting down on potential variables. But if I play my janky Jade Druid wannabe deck (as if I had the cards to actually make the meta deck...) the game goes to crap by turn 7 due to all the cards on the board. And if it's my Jany Jade versus a real Jade Druid, the game starts to stutter by turn 5 or earlier, depending on how many cards are on the board.
So far, the only way I have seen to get around this is to use the bot to hide the game window. While hidden the game continues to play and seems like it doesn't get bogged down, and can even fix some of the stuttering when you bring it back up. But I can't see the plays when this is happening, so I don't know for sure. When I hear the timer explode, I don't know if it's me or the other guy.
EDIT: Using the
unlock FPS edit posted above seemed to fix the issue for me. Going on 84 minutes with auto-playing an Aggro Druid (mostly) deck in Ranked and not detecting any stuttering.