As I mentioned earlier, this circuit switches the entire input power across the transformer primary when the transistor is in the ON state (555 pin 3 output High). The current is limited only by the internal resistances of the transformer primary and the Darlington transistor itself.
If you are trying to power this circuit with a 9V battery, it is very likely that the voltage is sagging so much that it just won't work.
Please place your voltmeter directly across the battery terminals, and then press the "on" button of your circuit. Do you see the voltage dropping?
Also, without knowing the precise details of the transformer you are using, there isn't any way to answer your question. No, the problem is not related to the "feedback", this is not that kind of circuit. If you post a photo of your transformer, and show where you are measuring the resistances and where you get "no resistance can be found" it may make it possible for us to help.
You may be interested in the attached scopeshots. This is the performance of my breadboarded version of your circuit. The Yellow trace is the secondary output of the transformer I am using, and the Blue trace is the Pin 3 output of the 555 timer. These traces are telling me that the oscillation frequency of the 555 is much too low for best performance from the transformer I'm using. However, even with only 9V input to the circuit, the transformer is still producing peaks of over 200 volts on the secondary output. But because the peaks are just a small part of each cycle and there is a lot of "dead time", there isn't really much power in the output.
You might try raising the frequency of your oscillator by lowering the value of the 10 nF capacitor and/or the 33k resistor. For example if you change the 33k to 2.2k the oscillator's frequency will go up to around 27 kHz. With my transformer this works a lot better. See the third scopetrace below. This is with 9V input from a regulated power supply, and is lighting up a NE-2 neon from the secondary.