It's hard to imagine a situation where property values rise would kick me out of my home, were Prop 13 repealed. If the home is going up massive in value, a 1-2% tax on the value of the home is dwarfed by the increase in property value. Refi or whatever to take care of it. The tax portion of a mortgage isn't that big anyway.
And if property values are rising so quickly that I even notice an increase in the property tax, then every single renter is completely screwed by comparison.
The only way that property value increases could cause somebody to lose their house is if they are completely financially incompetent, as far as I can see.