Unlike CLDR, IANA tzdata doesn't capture such names - other in comments. The tzdata name for Türkiye's time zone was and continues to be Asia/Istanbul.

On Wed, Sep 6, 2023, 9:50 AM Letu Yang via tz <tz@iana.org> wrote:

I understand this has been discussed last year, but since then both UN and US had adopted the name change from “Turkey” to “ Türkiye”.

Turkey's name changed to Türkiye
turkiye.un.org
For the State Department, Now It’s Türkiye, Not Turkey
nytimes.com

Recently Unicode CLDR data has updated the name in CLDR 43 as well.
Unicode CLDR - CLDR 43 Release Note
cldr.unicode.org

The Turkish government recently asked businesses operating in Türkiye to adopt the new name before the end of the 2023. I think it might be the time for the Turkey Time to be changed to Türkiye Time in tzdata.

Letu Yang