Property Taxes are terrible. It's the government forcibly making you a renter on land that YOU OWN or in most cases don't even own yet because you have a mortgage. If I didn't have to pay local property taxes I would have already paid off my house by now.
Taxes are the privilege of the sovereign. Who, for us, is the Federal Government of the United States, as well as the Government of the State in which we live, and the local government endowed with authority by the State in which we live. Taxes are levied as either fees for public services or as a tool for market intervention (e.g. property taxes for school districts, taxation on tobacco and alcohol). Ostensibly property taxes are used on services that make the area the property is in a better place to live, worth paying the tax for. Alternatively, property tax can be used as an intervention on the property market.
Abolishing property taxes outright would put more pressure on other sources of tax revenue (i.e. income and sales tax) in order to make up the funding without accompanying spending cuts. And as we should know by now, tax cuts usually do not come with accompanying spending cuts. So it's likely that tax burden would shift onto working taxpayers who don't own a home. Also the holding of property would be incentivized, since there would be decreased pressure to sell through property tax. Thus possibly raising house and land prices for a now more tax-burdened non-ownership cohort.
On the other side, abolishing property tax relief for seniors would increase pressure older owners to sell, or transfer ownership to their children. If your goal is to decrease the median age of homeownership, there's a decent chance it'd help. I'd say you shouldn't get tax assistance just because you're old. If the government claims tax on everyone's income or property, there should be no special treatment.
Taxes will always be stupid, retarded, and dumb. They will always be too high and the lot of it will spent on the wrong things. But they are the cost of Government. The same Government that's finally putting guns in the faces of illegal immigrants and dropping bombs on drug peddling spics in international waters. One step at a time.