1.I hate tipping. I hate it for the post-dinner math (that) it requires of me.

for介词短语表示hate的原因。it requires of me为定语从句，省略了作宾语的关系代词that/which. 比较：Tipping requires math of me. 给小费要求我做数学计算。

2.That's especially worrying, she says, given that anorexia results in more deaths than does any other mental illness, according to the National Institude of Mental Health.

than does any other mental illness = than any other mental illness does.

这是比较分句采用了倒装结构。当比较结构需要突出比较分句的主语时，则将主语后移到分句的句末（完全倒装），但分句的谓语必须为操作词或含有操作词（部分倒装的特点）。所以，比较分句的倒装为完全倒装和部分倒装的结合形式。例如：

Tom runs faster than his brother does/runs.

Tom runs faster than does his brother. 正确

Tom runs faster than runs his brother. 错误