I am an Indian citizen residing in Georgia with a temporary residence permit. Do I need a visa to visit Turkey?
Yes, you will need a visa. Turkey doesn't seem to care about your status in Georgia. Just in case you happen to have a visa from a Schengen country or an OECD country, you can use their e-Visa system which seems to be a quicker way to go about it.
India: Diplomatic passport holders are exempt from visa for their travels to Turkey up to 90 days. Ordinary, Special and Service passport holders are required to have visa to enter Turkey. Ordinary, Special and Service passport holders with a valid Schengen or OECD member's visa or residence permit may get their single entry e-Visas valid for one month via the website www.evisa.gov.tr, provided that they meet certain conditions.