Environmentally-friendly options are available for carpet underlay: felt, natural latex and jute.
See also carpet
If you have underlay in good condition, consider donating it or on-selling it. Smallish pieces (up to 2 metres) can be taken to the ReUse Academy.
Check with the manufacturer if there is a carpet-recycling programme which includes underlay.
If the underlay cannot be reused, take to the local waste transfer station.