What is the range of currents that I can safely draw out of a 9V battery without damaging it?
I am planning to design an experiment and I don't want to draw too much of current out of a battery and damage it.
This depends widely on the battery chemistry, size and manufacturer.
It is directly dependent to the serial resistance of the battery and the heat generated during discharge.
You should find the specific battery datasheet and look for the discharge current.
Some datasheet will give you a value in Amp, other will give you a value in C, like "2.5C" which refers to the battery Amp Capacity in A or mA. If the battery is 2500mAh, 2C would mean 5Amp, 0.5C would mean 1.25Amp.
For standard 9V batteries, this information is not always available, but these batteries are not designed to deliver high load.
This datasheet shows a discharge rate of 150mA continuous, at this rate the battery would be done in about 8 hours.
Given this is a high end battery, I would not get over 100mA discharge rate for a standard 9V battery.
Batteries are expendable components, therefore the fear of "damaging battery" doesn't sound well founded.
The range of current that can be delivered by a 9-V battery depends on its chemistry and quality of manufacturing (and design target). For example, a freshly made "Zeus 9V alkaline battery" can deliver 2.5 A of current (9.5V with 3.9 Ω load for 0.3s):
With voltage drop of just 0.1 V, this make the battery ESR of 0.1/2.5 = 0.04 Ω. SHorting this battery might create quite a peak of current.
The datasheet also shows "service output" for this battery under several scenarios. For example, you should be able to get 35 mA for 15-16 hours under continuous discharge (~550 mA-h capacity), and much longer with intermittent discharge.
So the "safe" range is about 1000 mA for 20-30 minutes, and longer with less load.