If you're using a revocable living trust, meaning you can change the trust and you keep possession of house that's inside the trust, then the local property tax assessor will ignore the trust and send you the annual property tax bill, while you're alive. So there's no property tax savings by putting your house into a trust.
When you sell the house, in this revocable living trust, (obviously before you die) the IRS will ignore the trust and you can use the $250k ($500k if married) home sale exemption just like you would if the house was not in a trust. (see https://www.carrolltaxfirm.com/post/income-tax-on-the-sale-of-your-home). So there's no income tax savings if you sell the home before you die with your house in a trust.
If the irrevocable living trust (now irrevocable because you're dead and can't change it) sells the house, then the IRS will tax the trust on the gain (the difference between the sale price and the house value on the date of your death). The IRS could instead tax the trust beneficiaries (perhaps your kids) if the trust is written to allow this. (see https://www.carrolltaxfirm.com/post/should-an-estate-asset-be-sold-after-it-is-distributed-to-estate-beneficiaries-or-before). So there's a potential larger income tax bill if the trust sells the home after you die and you have not written the trust properly.
If the trust distributes the house to its beneficiaries and they then sell the house, the IRS will tax the trust beneficiaries on the gain using, again, the difference between the sale price and the house value on the date of your death. So there's no income tax savings if the house is sold after your death by your trust beneficiaries.
The primary reason for placing your home into a trust is to allow the house ownership to transfer to your children without probate court approval.