Ooh, this seems fun. So I'm actually making a mod to D&D right now. Sufficed to say, the six main stats are in tact, but a couple have different names (Wisdom -> Perception, since it's more of an inherent attribute than a skill - picking up that thing you perceived is worth your attention is the skill. Constitution -> Vitality. I think that was it...), and each stat has four "sub-stats" that I won't get into in depth, but this could be relevant.
There are a couple things to be mindful of when talking about Strength. Primarily, Speed = Strength. Speed is a muscle moving quickly (you use the fast-twitch motor units). Power is just speed + Resistance. You can only accelerate so quickly when you're trying to move more mass. So a fast person is someone who has the most amount of strength with the least amount of SIZE. Of course, the issue there is that your strength is a measure of your muscular cross-section, so by definition a stronger person should be a more massive person. Strong people can be flexible, very coordinated, have great cardio, and can be QUITE FAST (they just tend to need more time to accelerate). The MAIN DOWNSIDE to gaining muscle is ENERGY DEMAND. As you become a more muscular person your Basal Metabolic Rate increases. Which is good for losing weight, but it also means you need a greater caloric intake or risk gradual atrophy. As well, each movement will likely require more energy, as it will need to move more mass. This is part of the reason why heavyweights exhaust before feather-weights. The amount of extra glycogen and phosphates the muscles can store is not sufficient to keep up with movement demands.
So Strength^ = A little less acceleration (though I don't know how much I'd calculate that), greater caloric requirement and more stamina demand throughout a physical task.
And the others?
Constitution is interesting. I haven't heard of someone with a particularly great immune system being immune to most pharmaceuticals or basic nutrition, but... maybe for funsies. Plus, Constitution CAN play a little tag-game with Perception (Touch), with sensitivity and all that. I dunno. Gonna leave CON alone here
If Dex includes Flexbility, then we must remind ourselves that being overly flexible can be detrimental - your muscles may be generally weaker, and lose some of their stretch reflex and thus making you more susceptible to injury (in extreme cases to a similar degree to inflexible folk). Other things usually covered by Dexterity - proprioception, coordination of movements, preventing unnecessary force or undue twitching - don't immediately seem to come into a downside zone as far as I can see.
Dex (Flex)^ = maybe less strength, maybe more prone to overstretching injuries (there should be a "strain" damage, like being put into an armbar or something)
Perception could definitely be bad if you have too much of it. Just general sensory-sensitivity. Bright lights, loud noises, offensive odours, bitter tastes and unpleasant tactile information (temperature, pressure, pain) could all be detrimental in certain situations. Perhaps you can play a balancing game between this and CON for Perception (Tactile), but I'm not sure how best to handle this.
Intelligence, now that's interesting. I love the idea of Smarter = More susceptible to Insanity (or even just pessimism). Maybe intelligence fights with Charisma, which I'll get more into there. I'm gonna reveal my big 4 for Intelligence:
Understanding (how well you can comprehend something presented to you. Your capacity to learn), Memory (self-explanatory), Reasoning (your ability to figure something out, often with limited information. Problem solving) and Cognition (the rate with which you can process. "Speed of thought"). That "Understanding" one, as well as "Reasoning" could DEFINITELY be an issue if you wanna play with Lovecraftian themes. Memory could be good bad or neutral depending on your personality. Cognition I don't necessarily see a downside to, but I'm sure someone can come up with something.
Int (understanding, reasoning)^ = More sanity damage?
And Charisma. So I'm gonna break again and reveal three of the Charisma stats: Influence (essentially your capacity to sway people. Force of personality), Will (sense of morale. Also effects spellcasting and mental defense), and Faith (not JUST in a deity; faith in yourself, faith in your friends, faith that things will turn out okay. Your ability to trust. Confidence). Will might not necessarily have a downside I can come up with, but Influence could be played with. Might think on it. Faith could definitely go wrong, as we have seen so often in real life. Too much Faith can turn into False Hope real fast.
Cha (Faith)^ = Zealotry? Overconfidence?
All in all I'd mostly rather have more of these things than less. I'm glad you brought this up though, this has been a fun thought experiment.