What are the consequences of the military coup in Turkey for travelers?

Update 19 July:

The FAA ban on flights from Turkey to US has been lifted. By now I assume that most flights to Turkey operate as scheduled, but it is best to check before flying.
You should still consider travel advice from your government. The Australian advice still is at the same level, with the threats now more due to terrorism than the attempted coup:

• The attempted coup in Turkey has failed and the situation is normalising. Localised military activity continues.
• Airport operations in Istanbul and Ankara are returning to normal, with additional security measures in place.​ ​Airports are very crowded. We recommend you confirm your travel arrangements before travelling to the airport.
• You should continue to avoid unnecessary travel, large gatherings and demonstrations.

and the latest update as of today:

Travellers are reminded of the high threat of terrorism in Turkey. We continue to receive information that indicates that terrorists may be planning attacks in Turkey which could take place at any time. Further attacks in urban centres are expected. It is likely some attacks will target western interests, including tourists. Due to the high threat of terrorist attack, we recommend you reconsider your need to travel to Ankara and Istanbul. Across Turkey, you should exercise heightened caution and be vigilant to your personal safety and surroundings. The level of our advice has not changed.

Update 16 July:
Current travel advice from the UK Foreign Office:

We strongly advise you to stay indoors, avoid public places, in particular demonstrations, and remain vigilant.

And similar, more detailed advice from Australia (applies to citizens of all countries IMHO, emphasis mine):

The security situation in Turkey, particularly in Ankara and Istanbul, remains tense. You should stay indoors and avoid unnecessary travel. The security situation may be fluid for some time. Ataturk international airport and Sabiha Gokcen international airport in Istanbul have resumed limited operations although delays and possibly flight diversions are likely. All inbound and outbound flights at Esenboga airport in Ankara have been cancelled until 1800 local time. You should contact your airline or travel agent for information about your travel arrangements, including connecting flights, but in the meantime suggest you avoid gathering at Ataturk Airport until confirmation of resumption of normal service. Avoid any large gatherings or demonstrations. The President has urged all citizens to show support for democracy on the streets in major cities on the evening of 16 July. Australians should avoid any such gathering. We continue to advise Australians to exercise a high degree of caution in Turkey overall and to reconsider their need to travel to Istanbul and Ankara. Higher levels apply in some, other parts of Turkey.

Many airlines have cancelled flights, e.g. Lufthansa

Lufthansa has cancelled all flights between Frankfurt and Istanbul, as well as, all flights between Munich and Turkey for today 16 July 2016 for security reasons. Our connections between Frankfurt and Antalya, as well as, the flight v.v. Frankfurt-Bodrum will be operated as planned.

US airlines are not allowed to fly in our out of Istanbul and Ankara airports as of yesterday:

Per the Federal Aviation Administration’s notice to airmen (NOTAM) dated July 15, 2016, U.S. airline carriers are prohibited from flying to or from Istanbul and Ankara airports. All airline carriers, regardless of country of registry, are prohibited from flying into the United States from Turkey either directly or via third country.

and see further travel advice from the US embassy in Turkey.

I do not have an overview but apparently some flights are going in and out of Istanbul again, mostly with Turkish Airlines. If you were scheduled to travel, contact your airline. Turkish has published a waiver for Refunding and Rebooking flights here.

Old post as of 15 July:
As usual flyertalk forums are the best source of info for flights into Istanbul. Check there for the latest info buried in discussions.

• Apparently as of now at least some flights of Turkish Airlines are still landing at IST while those of foreign carriers are diverted to other airports.
• BBC reports the airport (IST I assume) to be closed.

Travel advice for people in Turkey would be to stay put wherever you are and keep out of trouble. As of the Reuters report linked in the question:

The U.S. State Department said Americans in Turkey should shelter indoors.