Once a property’s appreciation exceeds $250,000 and begins building taxable appreciation, is it better to sell or hold?

Do the maths for the specifics of your situation in your (presumably US) tax environment. You could sell, crystalize your tax-exempt gains, and buy something new. You’d have various transaction costs – time & effort, change of tenant, 2 sets of re-decorating costs (one for your current property to help you get the best price, and one for the property you buy next, which is probably currently decorated to its owner’s tastes, not your tastes for what you want an investment property to look like), mortgage arrangement costs, legal fees. Here in the UK we also have a transaction tax to pay, based on the purchase price. There may be other details in other situation.

My initial guess, without having done a spreadsheet, is that the one-off costs of essentially flipping from one investment property to another will exceed the likely tax savings, on a possible future taxable sale, especially taking into account that you might decide never to sell the property so you never have that capital gains tax to pay.