It impacts the ability of buyers to qualify for a mortgage, and hence directly affects people's decisions on what/how much house to buy.
If you're in a HCOL area, that's probably irrelevant as buyers can often swing whatever (or pay cash). If L/MCOL, not so much. Some back-of-the-envelope numbers:
P&I on a $200k loan right now is hovering around $925. Figure insurance is $75 a month or something to make the math easy, so you're at an even $1000 before taxes. You could qualify for that (no property tax) loan with a gross income around $45k.
Now, if that place has $500/mo in property taxes, you're going to need more like $65k a year in income to qualify. That's a huge pool of eliminated potential buyers in a LCOL area and I'd fully expect it to affect prices and sales.
I think the bigger consideration is the cause of the tax increase. If money is just being shuffled around (ie, sales tax is being eliminated and replaced with higher property taxes) then not a huge issue. If the fire dept pension can't be paid and they have to crank up taxes every year and people are moving away (hello, all of Illinois)... run.
-W