Does NFC drain power if tag is attached constantly?
To open up my answer I will use information from the Android developer page, first sentence from sub heading Tag dispatch system:


  Android-powered devices are usually looking for NFC tags when the screen is unlocked, unless NFC is disabled in the device's Settings menu. 


This indicates that the device is using current to power the antenna continuously, waiting for a tag to come into the field, thus using small amounts of current, using &lt;100mA to power the TX circuitry for reading, when writing it uses more than 100mA in some instances. This information is from the PN532 datasheet, this is the short one and used in a few small open projects.

Another reference showing the low consumption of current ICs, an integrated bluetooth and NFC chip for these smart watches and health monitors, from the release article:


  The device achieves peak power consumption of 5.9mA for Bluetooth communication (@3.3V, -4dBm transmitter output power or receiver operation) and just 600µA or lower for NFC Tag communication (@3.3V).   


Reading through, cowboydan's link to the Application note, I wouldn't be surprised if that is implemented in some way on NFC phones, but I don't think you are going to get a designer from some well known company saying: 'Hey, here's the circuit and firmware we use with the NFC apps.' :) 

An NFC sensor sell sheet from AMS, the sensor is said to harvest 4mA at 3.3V typical, this will obviously have little affect onthe reader, also from reading the datasheets of common NFC tag type 2 chips the consumption is extremely low. 

The are several ways they may have implemented a method of not reading a Tag twice,


The firmware may store the UID from the Tag once it has performed its operation within the desired application and puts the reader into a sleep mode and waits x time, after which it restarts and reads the tag again and if it has changed, it runs the appropriate stuff again, else it waits again etc. Or
The firmware can read the tag and initiate the dispatch sequence and such used by the android apps, then store the Tag UID, and ignore that tag for a specific time before it sees it as another instance, without a sleep mode, or
The hardware and firmware implementation used on the Texas app note could be implemented to reduce current usage.


So to conclude the tags will vary with there current draw, 3&lt;mA&lt;30, these will not harvest much from the readers powering circuit when running at the typical ranges.

Also there will defiantly be firmware to 'silently' re-scan the tag to see if it is a different one, but this will probably be done so that it uses the least amount of power.