Not a rented house, but I can give you some idea of what you might need to consider.

Oil prices vary much more than electricity does. For example November last year the offer from Migrol I got was CHF105.40 per 100 litres for the standard oil. The last one I had from them in September was CHF86.40. Our tank holds 3,300 litres and we usually get it filled up when it drops below the 2,000 litre mark because that bring those prices into play. If I bought less it would be 3 francs more per 100 litres. Also check whether the oil can be bought via a bulk purchase with your neighbours, again that can lower the price because of the higher volume involved.

There'll be, or should be, an annual maintenance service for the boiler and also it will need to be inspected by a chimney sweep to make sure it's within the regulations. Note that the chimney sweep's word is effectively law, any problem they find must be fixed so they're happy with it. Whether you pay for these or the landlord does should be outlined in your rental contract.

Also check when the oil tank itself was last renovated, this has to be done every 10 years. I would think that should be a cost to the landlord, but again check your contract to be sure. What this entails is emptying the tank and cleaning all the muck that accumulates in it out then putting the oil back again.