The short answer is: no caffeine does not kill, interfere with, or otherwise affect D-chiro-inositol.
The term inositol is often used loosely when discussing myo-inositol, pinitol, etc, and this leads to some confusion. The only studies about inositol and caffeine are very specific to the effects of caffeine on the function of myo-inositol in the brain. There has never been a study about the effects of caffeine on D-chiro-inositol and insulin action.
However, we know that caffeine exacerbates the symptoms of PCOS. Caffeine acts on the liver, causing it to break down glycogen into glucose. The glucose is released into the blood, raising the blood sugar level. The pancreas, in turn, releases insulin in response to the blood sugar. The overall result is a series of wide swings in blood sugar level.
Women with PCOS typically have high blood sugar and insulin levels controlling these is a key to treating PCOS. This task is difficult enough without adding a wildcard like caffeine to the equation.
This is why we recommend against caffeine for women with PCOS. But if you choose to have that morning cup of joe, do so knowing that your DCI is not being destroyed.