My vitamin A and to a lesser extent my vitamin E tablets are starting to smell rank. I didn’t see anything on the bottle about them needing refrigeration after opening so I left them on the dresser instead of in the refrigerator.
Are you supposed to refrigerate them after you open them? What does this bad smell signify?
Well, they will last longer that way. IME, oil-soluble vitamins are delivered dissolved in oil, inside soft gel capsules. Given time, the oil can migrate through the caps, and will go rancid when it hits the air. What you’re smelling is probably rancid oil. I’ve never heard that rancid oils have any health effects, but I haven’t heard that they don’t, either.
Tablets??
If they really are tablets and not gelcaps, I’m sure they should be refrigerated. I don’t think I’d want to take them in the condition you describe. Not sure how the heat affects the vitamins themselves; I’ve no idea of the chemical formulas. I think we need a chemist. 
I buy gelcaps. They’re as cheap as any (Wallyworld), and they don’t go bad at room temperatures. OTOH, I live on the ground floor of a brick apt building, and even with lousy southern insulation, I can be gone all of a summer day, and my apt will still be in the 70s.
Just guessing … you turn off the AC when you’re not at home, and it gets pretty warm in there sometimes. Right? If so, when you do get replacements, keep them in the fridge anyway.