What does “Every Jewish mother will die, a fact many are not exactly famous for playing down” mean?

The phrase

Every Jewish mother will die, a fact many are not exactly famous for playing down

may be doubly confusing if you're not familiar with Jewish mothers and English is not your first language. There are two tricks going on here:

• There is a figure of speech here known as litotes. By denying that they play it down, the author is affirming that they play it up.
• The Jewish Mother stereotype has these women guilt-tripping, in this case by playing on their age and infirmity to get others (e.g., their sons) to pay attention to them (say, to visit).

a fact many are not exactly famous for playing down

Sarcastic understatement. The implication is that Jewish mothers put guilt trips on their kids with statements like "One day I'll be dead, then you'll wish you could pay me a visit!"

The "would it kill you..." is over the top. It is for someone who has not done what is expected of them. Wife to husband: "Would it kill you to take out the trash?!"