How can I make a fake fireplace hearth?

You could tile a box, but the cement would probably warp the cardboard. I would use aluminum or wood instead.

If you are wanting to sit on it make it so it won't crack. Build your wooden box out of 2 x4s and with 3/4" plywood and cover it with 1/2" concrete board. Make sure you have cross braces inside the box at least every 16",(Like floor joists) plywood and concrete board screwed around perimeter of box and on to all joists. Fill in joints and screw holes with tile morter. Install your tile, grout, and seal. Now you have a hearth.

Use tile mastic to adhere the tile. Thinset will not work.

Try covering the wooden box with cement backer board (like used in showers) before tiling the box. It would provide the appropriate surface for the thinset mortar as well as strengthen the hearth.

Be sure & post pictures of your finished project!

https://www.lowes.com/pd/DUROCK-Brand-0-5-in-x-36-in-x-60-in-Cement-Backer-Board/1000383889

https://www.lowes.com/pd/MAPEI-Floor-Tile-50-lb-Gray-Powder-Thinset-Mortar/3743787

You will need to add a layer of cement board over the the box. That will prevent the tile from cracking in the future and protect the wood.