Some of the specialized property tax reductions (like urban agriculture) can stay, but ideally we'll be pretty selective and get rid of the silly religious property tax exemptions.
What's your alternative? Become a slave for 30 years to maybe afford to buy a house?
If you could deduce rent (or mortgage for that matter) then more money would stay in the hands of ordinary people. It would also encourage cities to make housing affordable. If housing costs get too high then tax revenue would drop. I think that's a good incentive overall.
It is a good life choice to try and recoup some of the money you're paying into housing as an accrued asset, if you're currently renting I'd suggest looking at the numbers in greater detail - if you move frequently the tax your government levies on selling properties may tip this equation the other way but generally property ownership is a good idea.
To return to your original question, a subsidy on renting allows property owners to shift the price point on their rentals in their favor and ends up going into the owner's pockets, they can reinvest and grow their assets faster while people who are in the rental market will find it harder to escape due to an increased growth in property values - also a rental subsidy is great for anyone who already owns but I don't really think that's a section of the population we need to actively subsidize.
I bought my place for 150k back in 2002. I paid it off in less than 7 years, and have been mortgage free for over a decade now. I know another guy who did something similar, paying off his place is less than 6 years.
I realize this is very uncommon. Most people I know are perpetually refi-ing, and have no desire to ever pay off their mortgage.