Tomatoes are pretty high in acid and the stuff will keep for quite awhile. Make sure you’re not licking the spoon and putting it back in the jar, keep fingers away from the rim, etc. Some items like this keep better if stored upside down, for some reason.
My honey had a yeast infection once.
So let me get this straight: You want to keep something in the closet? 
I bet that made your biscuit taste funny.
It’s all about water activity, people, which is defined as the water vapor pressure over your condiment divided by the water vapor pressure over pure water. All of these substances have tons of nutrients that microbes would love to devor. Water is the limiting factor. And not just water, but free water, meaning water that’s available for them to use.
In mayo, for instance, there’s tons of water, but it’s all tied up in the emulsion, so bacteria are unable to grow. Even if the eggs that were used to make it were infected with Salmonella, it won’t be able to grow to dangerous levels. However, if you stick a knife in there with, say, a chunk of bread stuck on it, and the bread sticks in the mayo jar, there will be free water in the bread and lots of nutrients in the mayo, so the bacteria will go bonkers. Unless you refrigerate it, of course.
Honey is the same deal. The water is all tied up with the sugars. There are a few molds and yeasts that can grow in it if conditions are favorable, but it’s not common.
I don’t know the water activity of BBQ sauce off the top of my head (I know, I know. I’m sorry), but I’d hazard a guess that it’s along the same lines.
pH effects definitely help limit microbial growth, but water activity plays a vital role as well. One of these days, I’ll dig around and find my food microbiology textbook and provide some hard numbers.
Who the hell has a bottle of Hershey’s syrup long enough for it to go bad? What’s wrong with you people?