I have a 1950's oil run boiler system that needs to be replaced in a rental property that I own. Can you please tell me the approximate costs and benefits/energy savings expectations of least cost and highest energy saving options. Is there any tax, insurance or other savings/benefits to going with a higher energy saving replacement? Thank you.

If you pay the utility bills you will save some $$ on oil.
If the tenant pays the bill it might make the home more attractive to rent.
No insurance benefit that I know of.
For tax benefit check with your accountant because I do not know if the federal energy tax abatement applies to rental property.
No one can give you a approximate cost. The cost varies from town to town city to city depending upon material cost, unit size, actual additional material needed to do the job and local labor rate plus building permits.

Any unit you buy will be a big energy saver since your old unit is from the 50's.

If it was one of my rentals I would buy a new standard oil boiler and call it a day. Reason is that the real high efficiency boilers cost super big $$$$. They will need constant service also by a trained specialist who is used to working on that equipment. This is the 90-95% efficient oil (Monitor Products). You can have standard 80-84 % with less problems (Peerless and others).