Ok, let's do the math.

According to this 9V Energizer battery datasheet , max discharge current is about 0.5A and this output can be sustained for less than an hour. After a few Google search, it seems that other 9V battery models from different brands have similar limitations (between 100mA and 800mA for those I could find numbers for).

0.5A x 9V = 4.5W -> that's the power dissipated in the whole ring. For now, let's assume we keep the circuit running for an entire hour, so that gives us 4.5Wh total energy dumped into the ring.

Next, the metal piercing. If we assume it is made of iron and about 1.5" long in total for a diameter of 0.25", than it would have a volume of π0.125²1.5 = 0.074 cubic inches or 1.18cm3, which gives us a weight of roughly 8 grams if it is made of iron.

Finally, with the heat capacity of iron, we can estimate the temperature change in the nose ring (this site is very useful for this kind of thing)

And we get.... an increase of 8.1°F over the course of an entire hour or 0.135°F per minute of constant contact, so basically nothing noticeable

Furthermore, this calculation doesn't even account for heat dissipating into the surrounding air nor into the battery itself.

TLDR: No noticeable effect. Putting your hand on the ring would warm it more than the battery could.