No. No one in the public school system will care one bit how you did on the ISEE.
For the public schools:
The specialized schools will care about how you do on the SHSAT - all except Laguardia which cares only about whether you made their grade cut off and then about how you do on your audition.
The selective schools care about grades, mostly. But many selective schools have their own tests, interviews, portfolio evaluations, teacher recommendations, or any of the above.
The Ed. Opt. Schools care about how you ranked on your 7th grade ELA.
The limited unscreened schools care about whether you've expressed any interest in the school, beyond applying for it (attended an open house or tour usually.)
The zoned schools care about whether you live in the zone.
The other arts schools care about similar things to what Laguardia cares about. But Laguardia is MUCH harder to get into!