Do I need a visa to use a hotel facility offered by Turkish Airlines at IST for long layovers?

The free lodging rule of Turkish airlines is laid out on their site:

• A complimentary hotel accommodation will be provided to our valuable passengers when there are more than 10 hours (for economy cabin passengers) and 7 or more hours (for business cabin passengers) waiting during their international connecting flights.
• However, if your destination can be reached by an alternative flight to your first flight and the waiting time is less, you cannot take advantage of the hotel service.
• The first and the connecting flights must be with Turkish Airlines.
• All passengers must hold a confirmed reservation for the first and the connecting flights.

As you are flying to India, Turkish only has one service each to Delhi(DEL) and Mumbai(BOM) from IST so I expect you fulfil the "shortest" connecting time criteria and are therefore eligible for this scheme.

To avail this, the same link states

Complimentary accommodation will be arranged with Turkish Airlines’ corporate partners and will be provided through the “Hotel Desk” located after the customs’ check point at the arrival.

Thus to take advantage of this, you need to "enter" Turkey and require the visa. Note that you need a regular tourist visa.

Indian citizens can get a single entry eVisa provided they satisfy the following requirements:

• Passport covers the period of stay in Turkey.
• Return ticket, hotel reservation and at least 50 $ per each day of stay.
• valid Supporting Document (Valid visa OR valid residence permit from one of the Schengen Countries, USA, UK or Ireland).

If you fulfil ALL the above conditions, you can get the evisa online by paying 43$ fee. Otherwise you will need to apply at the Turkish embassy of the place where you are currently residing.

A good link for using this facilities (and explanation of its nitty gritty) is here