Usually, when we talk about gentrification, it is in terms of changes to the rental price as a result of changes in the local business environment, like firms opening and closing amd changes in wages; however, you are correct that the economics function roughly the same way when the government is raising property taxes to force a couple of empty nesters living off their 401k out of a 5 bedroom household to move to a 1 bedroom apartment as when an apartment complex is raising rents to force a young working family out of their two bedroom apartment to a less desirable neighborhood by raising the rent, especially because both can happen when a new park opens up down the street.

force poor owners out

This would increase the supply so it would reduce the value of the property and possibly rents because of that.

So it would harm poorer homeowners but may help renters, who tend to be even poorer.