If I refinance and my property appraises for more than when I bought the home, my property taxes will go up.
FALSE!
The property tax bill, because of Prop 13, can increase more than 2% of the assessed value year to year. The assessed value is determined by the County not an appraisal for a loan.
The appraisal for your loan is protected by privacy laws and therefore is not public knowledge and cannot be released to any outside entity without a subpoena.
So, if you bought your home in 2011 for $350,000 and now it appraises for $500,000 – CONGRATS! and relax. Your tax bill will remain the same or at least be protected by Prop 13 regulations and limits.