Almost certainly, yes. In the Church of England, any significant change to the interior of a church building requires a faculty, which is the church's equivalent of planning permission, granted by the Consistory Court on the advice of the Diocesan Advisory Committee.
Some minor heating works may fall under List B of the Faculty Jurisdiction Rules, which allows certain changes to proceed with archdeacon's approval rather than a full faculty. Replacing a like-for-like heater or adding a small portable unit might qualify. But installing a new heating system, changing fuel type, or mounting heaters on walls and ceilings in a listed building will almost always need a full faculty application.
The process starts with an informal conversation with your DAC secretary. Describe what you want to do, share your quote and any technical drawings, and ask for their advice before submitting the formal petition. This pre-application chat is free, saves time, and helps you avoid submitting a proposal that the committee will ask you to revise. Most DACs are genuinely helpful at this stage, so use them.