Will the new airmen certification standards make it more difficult to add ratings? As of June 15, 2016, the FAA has changed from PTS/Knowledge tests to the Airmen Certification Standards which is a relatively significant paradigm shift in how airmen are certified. 

The underlying reason to move to the ACS was to evolve the outdated PTS (practical test standards) and to align airmen certification towards risk management and the existing national airspace system.

The question I have though is for people who have earned their certificates under the old PTS system and would like to add a rating like an Instrument rating, or another certification like commercial.

Will the new ACS make it more difficult for existing pilots to add ratings? I'd like to know from the standpoint of part 141 and part 91 training centers. 

One of the reasons I ask is I went up with my CFI last week and went into some slow flight. I set up to bring the aircraft into the stall buffet and fly it a few knots above stall speed but he said that the new ACS said that the stall horn cannot sound during slow flight, which seems counter intuitive to me. The point of slow flight is to know how to handle the aircraft at or near stall. There is a lot of handling difference between VS0 and VS0+20kts. 

So that got me thinking, what would I have to relearn in pursuit of an instrument rating that would make it harder that I probably wouldn't have to under the old PTS?
Not significantly. The fundamentals of evaluation have not changed, and certainly the fundamentals of your next rating haven't changed. A few of the specific tasks have changed between PTS and ACS, like the point you raise. And, yes, you will need to review the ACS or PTS (whichever is in effect for your next rating -- not all have transitioned) in detail, so that you're aware of what's expected of you.

The major parts of any checkride will be your basic knowledge of the subject, your ability to move the aircraft through the air to standard, your ability to demonstrate sound judgment including a thorough evaluation of pre-flight risks and your overall attitude toward flying. That really hasn't changed with the ACS. The regulations have not changed either. The ACS is meant to give instructors and examiners a more holistic approach to evaluate the above factors.