What impact would falling oil prices have on the natural rate of unemployment? My intuition is that as the price level falls, cost of inputs is falling for suppliers and their output increases. More people are then employed to increase production - correct?
We need to distinguish between theory and the real world.

In theory, anything can happen. You would need a model that incorporates something that resembles an oil price (many standard models just have a single aggregated good). What then happens depends upon what other assumptions you stick into the model. Whether any of these models have any relationship to the real world is unclear. In any event, you would need to fix a particular model, and as a new question about its behaviour. (E.g., “What is the effect of an oil price rise in {Model X}?”)

In the real world, we need to distinguish the measured unemployment rate, and the “natural rate.” There are a number of ways of estimating the natural rate of unemployment, and what happens to them depends on all of the inputs into the estimation procedure.

What really matters in the real world is what happens to the measured unemployment rate (which may flow into the estimates of the “natural unemployment rate”, as many of the estimation procedures resemble low pass filters). I think the answer is a definite “it depends.”

Firstly, it depends on what country you are talking about. For an oil producer, falling oil prices can easily be a disaster for employment. Many oil producing countries have government revenues that are highly sensitive to oil prices, and so they may be forced to cut back spending. However, they generally have financial reserves, and so can handle limited price falls with limited disruption. Therefore, you need to look at each involved country separately.

For an oil consumer, a rapid rise in oil prices is disruptive; the 1970s experience could be used as an example. Consumers have to spend more on energy, making some products and services unviable. Meanwhile, the increased income is flowing to overseas producers. There is a “terms of trade” shock. The economy will eventually adjust to the new pattern of activity that incorporates higher oil prices, but in practice, the adjustment implies higher unemployment in the near run.

If the fall in oil prices is reversing a previous spike, this will presumably undo the previous disruption. However, if prices were previously stable, effects may be less dramatic. The effects on the economy will depend on how consumers react to the fall in energy prices. If they use the energy savings to buy domestically-produced goods and services, the total volume of domestic production rises, and so firms would presumably need to hire extra workers to produce the greater volume of output. However, the energy savings could just lead to higher savings, leading to no change in domestic production, and presumably employment (in the near run, at least).

The key is that falling input prices are generally not enough to cause increased production by itself; the firm needs to sell a greater quantity of goods, or else the extra production would just represent an undesired increase in inventories. (Since firms have to project demand, they could ramp up production (and hence, employment) to meet projected demand, but if that demand is not realised, they would have to reverse that decision.

For a country that is both a consumer and importer of oil (such as the United States) there is a trade-off between the benefit to consumers versus the potential loss of fixed investment in energy production. Therefore, it could go either way; you would need to do a fairly detailed model to get an estimate.