The taxes should indeed go up to optimize land use in general.
If you have a house built in 1902 in the middle of a big city, you're paying minimal taxes and find yourself surrounded by high-rises, it's very inefficient land use. Your lot could provide housing for hundreds if not thousands of people if another high-rise was built there.
Thus, you should either pay for your privilege (in taxes) or sell the lot (for a lot of money, presumably) to let other people use the space more efficiently.
However, I must say that a property tax itself is a wicked one and should be replaced by land value tax. Land is more valuable in the centre than in the outskirts of the city, so when properly taxed it creates the tendency to increase density of the highly taxed land and allows for more liberal land use where nobody else wants to build.
Property value goes up if you build a bigger, better building on your lot so you might as well stick with the old one. That promotes stagnation and under-use: it might not make sense for an individual or a business to give the lot to better use and move further away to enjoy lower taxes if the location is not particularly important.
In contrast, land value goes up in the average if a lot of people keep building desirable buildings in what turns out to be a desirable district in the city. That promotes what people like. Consequently, inefficient land uses move further away from the desirable areas.
Implementing neither property tax nor land-value tax is not straight-forward in general and there are corner-cases that need to be sorted, but in general land use is what ultimately provides its value. Often property tax codes already incorporate parts of land-value taxing, but the intrinsic value of property to be taxed is debatable: what is the value (to the community) of a small, low-taxed detached house in the downtown versus a huge, high-taxed mansion outside of the city? The opportunity cost of taxing property might be huge.
If you have a house built in 1902 in the middle of a big city, you're paying minimal taxes and find yourself surrounded by high-rises, it's very inefficient land use. Your lot could provide housing for hundreds if not thousands of people if another high-rise was built there.
Thus, you should either pay for your privilege (in taxes) or sell the lot (for a lot of money, presumably) to let other people use the space more efficiently.
However, I must say that a property tax itself is a wicked one and should be replaced by land value tax. Land is more valuable in the centre than in the outskirts of the city, so when properly taxed it creates the tendency to increase density of the highly taxed land and allows for more liberal land use where nobody else wants to build.