If by ‘save’ we mean cause less C02 emissions than an oil or gas boiler to generate the same amount of heat then the answer is an absolute yes.
Let’s look at the numbers – source is carbonindependent.org – other sources are available with slightly different figures – choose your favourite – it won’t change the answer by much!
In our case we moved from heating oil which emits 2.96 kg of CO2 per litre burnt and we used around 1600 litres of oil per year in our old boiler. That’s a total of 4.7 tonnes of CO2 per year.
The heat pump uses electricity and the UK grid emits around 0.309 kg of C02 for each kWh generated and delivered. Assuming an average co-efficient of performance (CoP) for the heat pump of 3.5 and using the house heat demand of demand of 12000kWh ( you can see this number on your EPC) per year that is:
0.309 kg of CO2/ kWh x 120000 kWh / 3.5 CoP = 980 kg CO2 per year
So around 1 tonne for the heat pump compared to 4.7 tonnes for our old oil boiler.
A well set up modern gas boiler would be around 2.7 tonnes for similar heat output – based on 0.203 kg CO2 per kWh.
CO2 emissions is a big and challenging subject – a good read if you want more is ‘How bad are bananas – The carbon footprint of everything’ by Mike Berners-Lee