I don't have any empirical evidence for this, but would say no for two reasons:
- An air filter will collect more dust, dirt, and debris until it gets so full that it starts pushing the older dust, dirt, and debris through the filter, which is what you are trying to prevent in the first place.
- As the air filter collects more dust, dirt, and debris, the filter element becomes more restricted causing less air to flow into the engine, hurting performance and fuel mileage.
A dirty filter will not make oil dirtier faster, but as it gets so full and the dust, dirt, and debris (hereafter called "stuff") starts pushing its way into the intake tract, it will cause buildup inside the engine. The "stuff" will get into the engine combustion chamber and create excess wear on all the moving parts. There is basically no way for the "stuff" to get into the oil. Oil gets dirty usually from cleaning excess hydrocarbons off of working parts and from the oil itself breaking down due to heat and age.
As far as the difference between new and old filter, a new filter will clean the air: an old filter won't do such a good job.