Learn Turkish: Tasarımda yakutlardan ve safirlerden kullanmak istiyorum. - I want to use some of the rubies and sapphires in the design.

This is the partitive use of the ablative case, which is the focus of this lesson! By saying yakutlardan (from the rubies), you indicate that you want to use some of a specific group of rubies. If you used the accusative case (yakutları), it would mean you want to use all of those specific rubies.

What if I just said yakut ve safir kullanmak istiyorum without any suffixes?

That would mean you want to use ruby and sapphire in a general, uncountable sense, as a material. Adding the plural and ablative suffixes (yakutlardan ve safirlerden) shows you are selecting a few individual stones from a specific collection you already have in mind.

Are the plural suffixes (-lar and -ler) required here for the 'some of' meaning to work?

For countable objects like jewels, yes. The partitive ablative acts like 'from among the...', so you need a plural group to take a portion from. That is why it is yakut-lar-dan (from among the rubies) and not just yakut-tan.

Why does the sentence use the locative suffix in tasarımda (in the design) instead of saying 'for the design'?

While you could say tasarım için (for the design), using the locative -da is very natural in Turkish here. It focuses on the physical placement of the stones, meaning they will physically be set in the final piece.