What does "Not recommended for new designs" mean in an ATtiny datasheet?

Not recommended for new designs means just that - they want to remove that SKU and are warning you that it will happen soon. If you are producing a large volume (often not actually that big, >10K), they might continue manufacturing for you on a contract basis.

This warning suggests you should move to a compatible or newer microcontroller for a design you want to start producing now and into the future. In this case, it would be the "ATTINY10-TSFR" if you want to have the same temperature range. If you can secure enough stock of the obsolete part for your purposes, then go for it, but be aware they might not be available in the future.

All electronic parts usually have a part status specified by the manufacturer, as follows:

• (Highly) Recommended. A new part that the manufacturer is pushing for and will keep producing for a long time ahead. Good to use in new designs.
• Active/Production. Normal status, meaning that the part is available, in production and good to use in new designs.
• Not recommended (for new designs). The part is still in full production but the manufacturer either offers a better equivalent part, or they plan to phase the part out of production. When they want to get rid of old parts they often keep raising the price.
• Obsolete. The part is no longer produced. Before a part goes obsolete, customers are often offered a Last time buy. A fixed last order date giving customers a chance to place one final order to cover their own needs for a foreseeable future.