There are no licensing requirements relating specifically to antique furniture restorers.
You should, however, be aware that if you spray solvent-based paints, lacquers and varnishes as part of your restoration activities then you may need an environmental permit (in England and Wales) or a pollution prevention and control permit (in Scotland and Northern Ireland). You can find out more about environmental and pollution control permitting on the Gov.uk website.
Note that premises in Wales which produce 500kg or more of hazardous waste in any 12 month period need to register with Natural Resources Wales.
The Export of Goods (Control) Order requires an export licence to be obtained for 'cultural goods' and antiquities over a certain age and value if they are to be exported from the UK. Full details are available from the Arts Council Export Licensing Unit. The Arts Council provides several guidance publications for exporters - you can find out more and download publications from the Arts Council website.
If you plan to work with certain restricted goods made from endangered species of plants or animals (for example ivory or tortoiseshell) then you should familiarise yourself with the CITES (Convention on International Trade in Endangered Species) licensing rules. Many restorers stick to using recycled and pre-Convention stocks of these materials.
Be aware too that there may be licensing issues relating to the possession - including temporary possession - of antique and historic firearms and weapons.