Is a 7 speed chainring compatible with an 8, 9 or 10 speed cassette and chain? Currently I have a 7 speed cassette (13-23) running with a spacer on a new Tiagra hub. After I changed my chain, the wear has made two of the gears unusable.

I am considering changing the cassette to increase the number of gears. The simple option would be 8 speed (12-23). The other alternative looking at the chainrings would be a complete change to 9 (11-23) or 10 (11-25) speed and drop a few teeth at the front.

Will my current chainring be incompatible with a new cassette and chain? In other words, can I use a 9 or 10 speed chain with a chainring designed for a 7 speed cassette and chain? Is there a significant difference in durability between the setups?

I mainly use by bike for commuting and light road touring.
7 speed systems use slightly wider chains, and the teeth on the chainring are slightly wider too. You might run into an issue with a ten speed chain not fitting over the teeth, or at least not fitting well.