Can I2C bus be left open circuit?
To answer the generic I2C aspect of the question, leaving the bus lines open isn't a good design practice because it'll pick up noise but I normally wouldn't expect any permanent damage. The main things I can think of that may happen are:


The clock and data lines will normally have a high impedance so leaving them open may pick up stray noise which may include things like mains hum, switching power supply noise and RF sources. Some combination of the above may cause the clock and data lines to be toggled quickly and randomly.
Transistors do dissipate more power the faster they are switched so I guess depending on the chip design there is some chance that a couple of internal transistors may overheat and fail, but I think the chance of the whole device releasing the "magic blue smoke" is extremely remote.
Because of all the random data potentially received on the clock and data lines there is a remote chance a device ends up in a state it's not designed for in that circuit. That might be something like an output gain block being set for too high for the application circuit, or a GPIO that defaults to an input ending up as an output.


Back to the tuner in particular I couldn't find a datasheet for the PLL used but looking at the block diagram and circuit configuration I couldn't really see anything where any of the above is likely. I'd put my bet on a general power rail connection / supply issue or that the tuner was defective to begin with.