Keep the plasterboard after removing the tiles?

That can be kept

It depends, if you are tiling a shower enclosure or where you are going to have a lot of water ( splash back for example) then I would recommend that you cover the plaster board with water resistant coating or replace the plasterboard with cement board ( either of these products are available from most builders merchant's or tiling shops .

Hi, I would personally suggest you should seal the area to stop water penetration onto the plasterboard as the could be hairline cracks as a result of the tiles being removed- replacing the plasterboard and then to have it skimmed is a costly business these days given the cost of materials. Please review the following product - Everbuild - Aquaseal Waterproof Tanking Membrane is a ready to use liquid waterproofing compound that is applied in a 2 coat system by brush. It is quick drying and tiling can commence after 12 hours. The tanking paint provides a permanently flexible watertight membrane on floors and walls prior to tiling in shower areas, wet rooms and wet zones and other areas subject to water exposure. It can be applied to all common building substrates such as plasterboard, MDF, plywood cementitious boards, blockwork, concrete & speciality wet room boards. Can be used on its own in many applications where a flexible watertight membrane is required. It can provide a 100% waterproof membrane to conventional plaster and timber based boards. Hope this helps you along your journey. Wass @ Line Design

Yes remove and install a new plasterboard, if is in the wet areas have to be tanking waterproofing, and after intall the tiles.

I would coat all plasterboard in tanking paint and as long as there is no mould on exsisting plasterboard then you can keep and tile on top