Felt under roof tiles is damaged

I'm presuming that you are talking about a sloping slate/tile roof not a flat roof. If the under felt is damaged it would be better to replace it but if the budget does not allow it you could make sure the sloping roof is not leaking and it is 100% watertight then fit foil back insulation between the rafters leaving a gap for an airflow above it & below it , then plasterboard over the rafters. There are huge numbers of roofs that are perfectly watertight that don't have any felt under the tile/slate covering.

Sorry but the only way to seal it is to remove the roof tiles,lathes & replace the felt, your more than welcome just to put plaster board up but remember to also but a bucket underneath. Good luck Alex