Your op amp does not have to be rail to rail.
In the resistive feedback loop to the negative terminal, the value of the gain is critical. If the gain is too low, the oscillations will dampen to zero. If it's too high, the oscillations will increase in amplitude until clipping and distortion. Also notice that, in order to start oscillating initially, the gain has to be high, so that some random fluctuation will be amplified until it results in full scale oscillation. When oscillation is established, the gain needs to be reduced in order to keep the amplitude steady and low in distortion. To make a good sine wave oscillator with low distortion and a steady amplitude, some sort of active adjustment to the gain in that resistor network is usually used. The classic approach is to use a light bulb, which increases its resistance as more current goes through it. That's what Hewlett Packard did for their classic Wien Bridge audio oscillator. There are other possible approaches, using PTC thermistors, diodes, and/or transistors.
If you're not seeing any oscillation, then I'd guess your resistor values in the negative feedback side might be resulting in insufficient gain. Of course, the feedback on the positive side also has to be correct...
Here are a few info sources:
https://www.calvin.edu/~pribeiro/courses/engr332/Handouts/oscillators.pdfhttp://www.zen22142.zen.co.uk/Design/wien_osc/Wien-Bridge%20Oscillator.htm