Sale of rental home after taking out equity

No. But the interest on the new loan is probably not deductible.

Removing equity via a refinance does NOT provide additional basis; it does NOT reduce the gain.

And as Susan said, if the cash withdrawn isn't used for improvements to the property, the interest is (with few exceptions) no longer deductible.

Payment of debt does not reduce the gain. See others for more explanations.

Borrowing against it wouldnt reduce his taxable gain at all.

Youre still left with purchase price plus improvements vs. sales price less expenses and then youve got the depreciation to recapture as well.

What you owe on the mortgage doesnt factor into the taxability of the sale.

You and your client are confusing Equity (or worth minus loan balance) and Gain.

Equity = your ownership position. The more in debt you are against the value of the property, the less Equity you have in it. That means it is more leveraged than if there was no debt on it. That doesn't have anything to do with Basis or Gain. You could buy something for cash or fully as debt, and your Basis is the same. Your Equity is different, though.

sjrcpa tells you how to understand Basis.

Gain (simplified) is:

Sales Price minus Basis (and further reduction for allowed costs of making that sale happen, such as commissions paid) = Gain or Loss.

It seems both you and your client need some better guidance on this.