I sold my late mother's property a few years ago.
The buyer asked me to get the boiler serviced and I said no, mainly because I knew the boiler was many years old (more than 25 years), in good working order and that my mum had always had it serviced.
The boiler was left on during the property sale. I couldn't find the latest boiler service so sent a copy of the last one I could find, which was two years old. I knew it wasn't a legal requirement and that the buyer couldn't force me to service it. The buyer wasn't satisfied so he asked if he could commission his own gas engineer to inspect it, which I agreed to do.
There were a couple of issues with the boiler and the buyer wanted to renegotiate the price, however as the boiler was deemed to be in good working order and safe, I held firm on the price and the buyer decided to carry on with the purchase, thank goodness.
OP, the seller doesn't have to get the boiler serviced for you, but I would be wary of the seller not allowing you to pay for it yourself. I would explain that I just want to be sure that there are no poisonous fumes escaping from the boiler which could be a risk to my family's health, but that I am committed to the purchase - just want to be sure that it is safe to use, particularly as Winter is around the corner.
Bear in mind that a boiler can break down the day after you move in, even if a service a couple of weeks before didn't reveal anything to be alarmed about.