I have created some shops, to sell items such as TVs or decorative plants. When the sim that owns the lot plays it, everything is fine. But when I visit the shop with some other sim, I can't buy anything. The active sim grabs the item, goes to pay for it... and the owner stays idle, not doing anything, ignoring the customer that wants to pay. Same thing happens if my active sim is simply there, and it is some townie who tries to buy something.

I tried removing all the shop-related hacks, but the problem persists. What else can I do?

This is a design flaw in the game itself. Either:

1. Make the owner temporarily selectable (the Blender or Inge's cat teleporter can do this, as well as good old testingcheatsenabled) and send them to ring up the customers.
2. Get Pescado's businessrunsyou. Get the big hammer and sickle and stick it on the wall (I usually put it in a bathroom). There's an option for uncontrolled owner - set that to cashier.

It can take an owner a long time to get to the till, it works best if you hire somebody and they are assigned to work the till, because the owner isn't technically assigned to anything. It's not a mod conflict it's just annoying vanilla game behavior. If you wait long enough they should come over to ring them up at some point. I find it easier to either hire somebody (hire family members mod allows you to hire and assign sims from the same house) or use an automatic till or what ihate suggested. In some of my shops I click on the sim, make them selectable and tell them to go ring up the customer.

I don't have that issue and I don't have Business Runs You or any OFB facilitator as far as I know. So I think you should test by removing more than your shop-related hacks.
Like Jo said, you might have to wait a little if the owner is busy elsewhere, restocking usually, but the fact that your owner stands idle sounds fishy.

I have the same problem, and I'm sure it doesn't have to do with mods. In my case, the owner wasn't even restocking stuff or anything, just staring at things and casually talking with people. I haven't tried it yet, but this looks promising to me