You can't reasonably create a shelf-stable lime juice and oil dressing in a home kitchen.
Oil and vinegar dressings don't really need much to preserve them because both oil and vinegar are shelf-stable on their own.
Lime juice, however, degrades at room temperature, so you would need to sterilize it in a hot water bath. That's pretty simple; lime juice is extremely high acid, so 10 minutes at 180F will do it.
However, that brings up a second problem: hot water bath canning anything containing a lot of oil is not recommended. There's some problems with canning anything containing a lot of oil:
- Any oil on the jar seal can cause it to fail;
- Oil can go rancid in the jar, even if heat-treated and sealed;
- Times and temperature formulas for non-oil packing do not work for oil packing, because oil takes longer to transmit heat;
- Adding the oil lowers the overall acidity of your dressing, possibly making it a low-acid food and requiring pressure canning.
Moreover, because home canning with oil is officially not recommended, it's very hard to find any reputable advice on how to do it. This means you'd be experimenting ... not just with your recipes, but with the health of your customers, which is very risky in most places.
However, there is another option. Many small producers sell dressings that need to be refrigerated and consumed within a few weeks; I have a feta-cheese based dressing from a local cheesemaker in my fridge right now. You could experiment in your own fridge to see how long it keeps. Consider this as your path forwards for selling your dressing.